diff --git a/diffeomorphic_approx_3d_lp_le2_v2.txt b/diffeomorphic_approx_3d_lp_le2_v2.txt
index 5605a24..ee0a900 100644
--- a/diffeomorphic_approx_3d_lp_le2_v2.txt
+++ b/diffeomorphic_approx_3d_lp_le2_v2.txt
@@ -142,7 +142,7 @@ hypothesis on the weak derivative.
 
 A technical convenience used repeatedly is this: if H : U -> V is a
 homeomorphism and a continuous map G : U -> V satisfies
-|G(x) - H(x)| < (1/2) dist(H(x), R^n \ V), then the straight-line
+|G(x) - H(x)| < (1/2) dist(H(x), R^n minus V), then the straight-line
 homotopy from H to G stays inside V; if V is bounded the homotopy is
 proper. This will always be combined with local injectivity and a
 degree argument (Lemma 5.3); it is never used as a substitute for
@@ -367,7 +367,7 @@ of f(K) in N, delta-balls may be chosen so that f_1 is still onto N.
 Proof. Transport the PL structure of M to N by f. Under this transport
 f is the identity, and K is a subpolyhedron on which the identity is
 already PL. Hamilton's relative uniform approximation theorem applied
-to the identity map of the open set M \ K (with target the enlarged
+to the identity map of the open set M minus K (with target the enlarged
 N) gives, locally on compact sets, PL maps that agree with the identity
 in a compact neighborhood of K and are arbitrarily close to it in the
 uniform metric. Local finiteness of the needed compact exhaustion and
@@ -391,7 +391,7 @@ embedding,
 then there is a homeomorphism A : R^3 -> R^3 such that
 
         A = e   on C,
-        A = id  on D union (R^3 \ int B),
+        A = id  on D union (R^3 minus int B),
         sup_{R^3} |A - id| < lambda.
 
 The number delta is uniform over any finite list of fixed normalized
@@ -433,13 +433,13 @@ LEMMA 5.3 (fine control and the target). Let h : Omega -> Omega' be an
 orientation-preserving homeomorphism between bounded domains of R^3,
 and let g : Omega -> R^3 be a smooth map with det Dg > 0. If
 
-        |g(x) - h(x)| < (1/2) dist( h(x), R^3 \ Omega' )   (x in Omega),
+        |g(x) - h(x)| < (1/2) dist( h(x), R^3 minus Omega' )   (x in Omega),
 
 then g is a diffeomorphism of Omega onto Omega'.
 
 Proof. Put
 
-        d(y) = dist( y, R^3 \ Omega' ),    g_t = (1 - t) h + t g.
+        d(y) = dist( y, R^3 minus Omega' ),    g_t = (1 - t) h + t g.
 
 Condition (1) puts the whole segment g_t(x) in Omega'. d is 1-Lipschitz,
 so
@@ -704,7 +704,7 @@ unit ball and set
 
 for x in Omega and w : Omega -> (0, infinity) a smooth positive
 function so small that x + w(x) z stays inside Omega for every z in
-supp rho. This condition is implied by 0 < w(x) < dist(x, R^3 \ Omega)
+supp rho. This condition is implied by 0 < w(x) < dist(x, R^3 minus Omega)
 near the boundary.
 
 Differentiating under the integral,
@@ -814,7 +814,7 @@ This is smooth. Indeed: on the annulus e^{-L_0} R_v <= r <= R_v, the
 map (r, theta, tau) |-> F_tau(r theta) is jointly smooth; substituting
 tau = eta(r) gives a smooth composition. At r = R_v both expressions
 equal F(x); if you insist, multiply eta by a further cutoff that
-vanishes to all orders at r = R_v so that the transition is C^\infty
+vanishes to all orders at r = R_v so that the transition is C^infinity
 across the interface. At the center, for r <= R_v e^{-L_0}, the map
 equals F_1(r theta) = r c q0(theta) = c (q0 as a rotation matrix)(x),
 which extends smoothly to 0.
@@ -872,10 +872,10 @@ At every stage:
 
 So g : Omega -> Omega' is a smooth map with det Dg > 0 everywhere on
 Omega, and || g - h ||_{W^{1,p}} < eps, |g - h| < xi, where xi is
-chosen smaller than (1/2) dist(h(x), R^3 \ Omega').
+chosen smaller than (1/2) dist(h(x), R^3 minus Omega').
 
 By Lemma 5.3, g is a diffeomorphism of Omega onto Omega', with
-C^\infty inverse. QED (Proposition 6.1, and therefore Theorem C, since
+C^infinity inverse. QED (Proposition 6.1, and therefore Theorem C, since
 The next part supplies the reduction from locally bi-Lipschitz H to a
 locally finite PL homeomorphism h.)
 
@@ -944,7 +944,7 @@ LEMMA 7.1 (smooth local tolerances). Let U subset R^n be open and let
 a : U -> (0, infinity) be continuous. For every L > 0 there is a
 positive L-Lipschitz function d on U with
 
-        d(x) <= min{ a(x), 1, L dist(x, R^n \ U) }.
+        d(x) <= min{ a(x), 1, L dist(x, R^n minus U) }.
 
 There are also positive smooth minorants satisfying prescribed
 positive continuous upper bounds on values and first derivatives. One
@@ -953,7 +953,7 @@ family of compact regional tests. Finite collections of such
 requirements can be combined, and error budgets on a countable compact
 exhaustion can be made summable.
 
-Proof. Extend e(x) = min{a(x), 1, L dist(x, R^n \ U)} by zero outside
+Proof. Extend e(x) = min{a(x), 1, L dist(x, R^n minus U)} by zero outside
 U, and put
 
         d(x) = inf_{y in R^n} { e(y) + L|x - y| }.
@@ -1013,7 +1013,7 @@ The upper bounds on the fine scale remain arbitrarily prescribable.
 
 Proof. Choose a positive size function s on Omega with sufficiently
 small absolute Lipschitz constant, below all the prescribed local
-bounds and below 10^{-3} dist(x, R^3 \ Omega). Lemma 7.1 supplies such
+bounds and below 10^{-3} dist(x, R^3 minus Omega). Lemma 7.1 supplies such
 a minorant. Take the maximal dyadic cubes for which
 
         l_Q <= inf_Q s.
@@ -1436,7 +1436,7 @@ mollification is done so that
         int_{Q_delta} sqrt(det g_0) <= A(h) + eps/3.                 (8.3)
 
 Extend g_0 smoothly to a positive-definite g on all of Q by blending,
-via a smooth radial cutoff supported in the shell Q \ Q_delta, to the
+via a smooth radial cutoff supported in the shell Q minus Q_delta, to the
 constant scalar M^2 I. The number M > 0 is fixed by the size of g_0 on
 the shell (finite because h is locally bi-Lipschitz). On a two-sided
 neighborhood of bdry Q the metric g equals M^2 I exactly.
@@ -1456,29 +1456,29 @@ reflection across the two perpendicular sides at each corner further
 extends Phi past each corner. Consequently Phi and Phi^{-1} are
 bi-Lipschitz up to bdry Q.
 
-Stage C (Cauchy-Schwarz on R). Let \hat h = h o Phi^{-1} : R -> R^d.
-The two coordinate functions of \hat h are \hat X, \hat Y : R -> R.
+Stage C (Cauchy-Schwarz on R). Let h-hat = h o Phi^{-1} : R -> R^d.
+The two coordinate functions of h-hat are X-hat, Y-hat : R -> R.
 Because Phi is conformal from (Q, g) to (R, Euc), and by the conformal
 invariance of the Dirichlet energy in 2D,
 
-        int_R | D \hat h |^2_{Euc} dy = 2 A(h).                       (*)
+        int_R | D h-hat |^2_{Euc} dy = 2 A(h).                       (*)
 
-The boundary values of \hat X and \hat Y are the coordinate pullbacks:
-\hat X(0, .) = 0, \hat X(W, .) = S; \hat Y(., 0) = 0, \hat Y(., S) = S.
+The boundary values of X-hat and Y-hat are the coordinate pullbacks:
+X-hat(0, .) = 0, X-hat(W, .) = S; Y-hat(., 0) = 0, Y-hat(., S) = S.
 Hence
 
-        int_R partial_1 \hat X dy = S^2,
-        int_R partial_2 \hat Y dy = S W.
+        int_R partial_1 X-hat dy = S^2,
+        int_R partial_2 Y-hat dy = S W.
 
 Expanding,
 
-        int_R (|partial_1 \hat X - S/W|^2 + |partial_2 \hat X|^2)
-          = int_R |D \hat X|^2 - 2 (S/W) int_R partial_1 \hat X
+        int_R (|partial_1 X-hat - S/W|^2 + |partial_2 X-hat|^2)
+          = int_R |D X-hat|^2 - 2 (S/W) int_R partial_1 X-hat
                         + (S/W)^2 (W S)
-          = int_R |D \hat X|^2 - S^3/W  =: Defect_X  >= 0,            (8.4)
+          = int_R |D X-hat|^2 - S^3/W  =: Defect_X  >= 0,            (8.4)
 
-        int_R (|partial_1 \hat Y|^2 + |partial_2 \hat Y - 1|^2)
-          = int_R |D \hat Y|^2 - S W    =: Defect_Y  >= 0.            (8.5)
+        int_R (|partial_1 Y-hat|^2 + |partial_2 Y-hat - 1|^2)
+          = int_R |D Y-hat|^2 - S W    =: Defect_Y  >= 0.            (8.5)
 
 Adding and using (*):
 
@@ -1489,19 +1489,19 @@ Also S^3/W + S W = S (S^2/W + W) >= 2 S^2 by AM-GM, so
         A(h) >= S^2.                                                  (8.7)
 
 Sharp case. Suppose A(h_j) <= 1 + o(1) and h_j|_{bdry Q} -> iota|_{bdry
-Q} uniformly. The image \hat h_j(R_j) then converges to the unit square
+Q} uniformly. The image h-hat_j(R_j) then converges to the unit square
 in R^2 x {0}, so W_j S_j -> 1 and W_j, S_j -> 1. By (8.7) this forces S_j
 -> 1, W_j -> 1. Then (8.6) gives Defect_X + Defect_Y -> 0, i.e.
 
-        partial_1 \hat X_j -> 1,  partial_2 \hat X_j -> 0,
-        partial_1 \hat Y_j -> 0,  partial_2 \hat Y_j -> 1     in L^2(R_j).
+        partial_1 X-hat_j -> 1,  partial_2 X-hat_j -> 0,
+        partial_1 Y-hat_j -> 0,  partial_2 Y-hat_j -> 1     in L^2(R_j).
 
 After the rescaling (y, t) -> (y/W_j, t/S_j) of R_j to Q, which preserves
 L^2 energy up to 1 + o(1), giving (8.2) with rho_j = Phi_j^{-1}
 followed by the rescaling.
 
 Weak case (8.1). If boundary closeness is not assumed, one no longer
-has S_j, W_j -> 1. Blend rho on the outer shell Q \ Q_delta to the
+has S_j, W_j -> 1. Blend rho on the outer shell Q minus Q_delta to the
 identity, with a cutoff going from rho at the inner boundary of the
 shell to the identity at bdry Q. The tangential derivative of the
 radial schedule is arranged not to interfere with D_t h. After the
@@ -1742,7 +1742,7 @@ extends as a homeomorphism from boundary B_i to boundary B_o by
 coning across the planar annulus between the two equators of the two
 spheres. The two half-extensions (across the equatorial plane) agree
 on the planar annulus, giving an ambient homeomorphism of
-B_o \ B_i that fixes the boundary of B_o and equals A on B_i. Choose
+B_o minus B_i that fixes the boundary of B_o and equals A on B_i. Choose
 a smooth cutoff supported near B_o; extend by the identity. In the
 inverse homeomorphism, the inverse affine germ is affine again. Both
 supports can be made small because h and h^{-1} are continuous.
@@ -1769,7 +1769,7 @@ The injective-amalgam form of Seifert-van Kampen then embeds each
 piece's pi_1 in pi_1(X) = Z. The common pi_1 already maps onto Z, so
 each piece inclusion is an isomorphism onto Z.
 
-Combining with the standard product structure of S^3 \ K, both pieces
+Combining with the standard product structure of S^3 minus K, both pieces
 of X are one-holed thickened spheres with a straight arc (the proper
 interval K intersect B) running through. Any such structure is
 topologically standard by Brown-Schurle-type regular neighborhood
@@ -1792,10 +1792,10 @@ from boundary^+, with boundary circles on boundary^- union boundary^+
 each meeting one L_i exactly once, transversely. Suppose the induced
 isomorphism
 
-        pi_1(F_1) -> pi_1(M \ S)
+        pi_1(F_1) -> pi_1(M minus S)
 
 preserves the individual peripheral conjugacy classes of boundary
-circles. Then (M, S) is the standard product M \ S = P x [a, b] with
+circles. Then (M, S) is the standard product M minus S = P x [a, b] with
 P = S^2 minus n marked points and S = {marked points} x [a, b], and
 the stated boundary parametrization identifies product ends.
 
@@ -1830,11 +1830,11 @@ that system exactly once, with the arc passing from one side of the
 sphere to the other. Impose one of the following angular tests.
 
         Test 1. Finite based generating loop system (gamma_1, ...,
-        gamma_{n-1}) for S^2 \ D, chosen outside the closed angular
+        gamma_{n-1}) for S^2 minus D, chosen outside the closed angular
         caps of all cones. At arbitrarily small common source radii r,
         the parametrized loops s |-> h(r gamma_a(s)) /
         |h(r gamma_a(s))| are all close to gamma_a within a fixed
-        simultaneous homotopy tolerance in S^2 \ D; the tolerance
+        simultaneous homotopy tolerance in S^2 minus D; the tolerance
         keeps image loops outside cone caps.
 
         Test 2. Only forward directions, cyclically ordered on a
@@ -2679,7 +2679,7 @@ and the center draw is equivalent, up to fixed bounded densities,
 to integration in the actual center a and normalized relative
 offset variables. The weight in a block sum is h^m. Failure to
 choose a in K on candidate blocks costs at most
-C |N_{Ch}(K) \ K|, which tends to zero.
+C |N_{Ch}(K) minus K|, which tends to zero.
 
 For almost every center in K and almost every fixed choice of
 relative variables, each of the finitely many forward line traces
@@ -2784,7 +2784,7 @@ fix a matrix D_1 (the local linear model of Df), with D_1 of
 constant rank k in {1, 2, 3}. Refine a Whitney decomposition into
 closed pieces so that f |_K_1 is C^1 and Df |_K_1 is within delta
 of D_1, uniformly. Discard the small leftover integral
-int_{near K_1 \ K_1} |Df - D_1|^p.
+int_{near K_1 minus K_1} |Df - D_1|^p.
 
 For k = 3, D_1 has finite inverse; a compatible first jet extends to
 a local C^1 diffeomorphism b (the radial-block construction applies directly).