let n be Nat; :: thesis:
let E be compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis:
let m, j be Nat; :: thesis:
set a = N-bound E;
set s = S-bound E;
set w = W-bound E;
set e = E-bound E;
set G = Gauge (E,n);
set M = Gauge (E,m);
set sn = Center (Gauge (E,n));
set sm = Center (Gauge (E,m));
assume that
A1: 1 <= m and
A2: m <= n and
A3: 1 <= j and
A4: j <= width (Gauge (E,n)) ; :: thesis:
A5: width (Gauge (E,m)) = len (Gauge (E,m)) by JORDAN8:def 1
.= (2 |^ m) + 3 by JORDAN8:def 1 ;
A6: width (Gauge (E,n)) = len (Gauge (E,n)) by JORDAN8:def 1
.= (2 |^ n) + 3 by JORDAN8:def 1 ;

A7: now :: thesis:

let t be Nat; :: thesis:
assume that
A8: width (Gauge (E,n)) >= t and
A9: t >= j ; :: thesis:
A10: len (Gauge (E,m)) = width (Gauge (E,m)) by JORDAN8:def 1;
A11: len (Gauge (E,n)) = width (Gauge (E,n)) by JORDAN8:def 1;
A12: 0 < (N-bound E) - (S-bound E) by SPRECT_1:32, XREAL_1:50;
A13: t >= 1 by A3, A9, XXREAL_0:2;
A14: 0 < 2 |^ m by NEWTON:83;
A15: 1 <= len (Gauge (E,m)) by GOBRD11:34;
then A16: ((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))) `1 = ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `1 by A1, A2, A8, A10, A11, A13, JORDAN1A:36;
A17: ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `1 = ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `1 by A1, A2, A3, A4, A8, A11, A13, JORDAN1A:36;
[(Center (Gauge (E,n))),t] in Indices (Gauge (E,n)) by A8, A13, Lm1;
then A18: ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2 = |[((W-bound E) + ((((E-bound E) - (W-bound E)) / (2 |^ n)) * ((Center (Gauge (E,n))) - 2))),((S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ n)) * (t - 2)))]| `2 by JORDAN8:def 1
.= (S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ n)) * (t - 2)) ;
[(Center (Gauge (E,m))),(width (Gauge (E,m)))] in Indices (Gauge (E,m)) by A10, A15, Lm1;
then A19: ((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))) `2 = (S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ m)) * ((width (Gauge (E,m))) - 2)) by Lm2;
A20: ((2 |^ m) + 1) / (2 |^ m) >= ((2 |^ n) + 1) / (2 |^ n) by A2, A14, Th1, PREPOWER:93;
t - 2 <= ((2 |^ n) + 3) - 2 by A6, A8, XREAL_1:9;
then (t - 2) / (2 |^ n) <= ((2 |^ n) + 1) / (2 |^ n) by XREAL_1:72;
then (t - 2) / (2 |^ n) <= ((width (Gauge (E,m))) - 2) / (2 |^ m) by A5, A20, XXREAL_0:2;
then ((N-bound E) - (S-bound E)) * ((t - 2) / (2 |^ n)) <= ((N-bound E) - (S-bound E)) * (((width (Gauge (E,m))) - 2) / (2 |^ m)) by A12, XREAL_1:64;
then A21: (S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ m)) * ((width (Gauge (E,m))) - 2)) >= (S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ n)) * (t - 2)) by XREAL_1:6;
A22: 1 <= Center (Gauge (E,n)) by JORDAN1B:11;
Center (Gauge (E,n)) <= len (Gauge (E,n)) by JORDAN1B:13;
then ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2 >= ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `2 by A3, A8, A9, A22, SPRECT_3:12;
hence (Gauge (E,n)) * ((Center (Gauge (E,n))),t) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) by A16, A17, A18, A19, A21, GOBOARD7:7; :: thesis:

end;

then A23: (Gauge (E,n)) * ((Center (Gauge (E,n))),(width (Gauge (E,n)))) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) by A4;
(Gauge (E,n)) * ((Center (Gauge (E,n))),j) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) by A4, A7;
hence LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),(width (Gauge (E,n))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) by A23, TOPREAL1:6; :: thesis: