let m, n, j be Element of NAT ; :: thesis: for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= m & m <= n & 1 <= j & j <= width (Gauge (E,n)) holds
LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j)))
let E be compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ( 1 <= m & m <= n & 1 <= j & j <= width (Gauge (E,n)) implies LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) )
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: LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j)))

now

A5: 0 < (N-bound E) - (S-bound E) by SPRECT_1:32, XREAL_1:50;
then A6: (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ m)) <= (S-bound E) - 0 by XREAL_1:13;
A7: ((N-bound E) - (S-bound E)) / (2 |^ n) <= ((N-bound E) - (S-bound E)) / (2 |^ m) by A2, A5, Lm7;
A8: 1 <= len (Gauge (E,m)) by GOBRD11:34;
then [(Center (Gauge (E,m))),1] in Indices (Gauge (E,m)) by Lm4;
then A9: ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 = (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ m)) by Lm11;
let t be Element of NAT ; :: thesis: ( 1 <= t & t <= j implies (Gauge (E,n)) * ((Center (Gauge (E,n))),t) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) )
assume that
A10: 1 <= t and
A11: t <= j ; :: thesis: (Gauge (E,n)) * ((Center (Gauge (E,n))),t) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j)))
( 1 <= Center (Gauge (E,n)) & Center (Gauge (E,n)) <= len (Gauge (E,n)) ) by Lm3;
then A12: ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `2 by A4, A10, A11, SPRECT_3:12;
A13: len (Gauge (E,n)) = width (Gauge (E,n)) by JORDAN8:def 1;
then A14: t <= len (Gauge (E,n)) by A4, A11, XXREAL_0:2;
then A15: ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `1 = ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `1 by A1, A2, A10, A8, Th57;
A16: [(Center (Gauge (E,n))),t] in Indices (Gauge (E,n)) by A10, A14, Lm4;
then A17: ((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)) by EUCLID:52 ;

A18: now

per cases ( t = 1 or t > 1 ) by A10, XXREAL_0:1;

suppose t = 1 ; :: thesis: ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2

then ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2 = (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ n)) by A16, Lm11;
hence ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2 by A7, A9, XREAL_1:13; :: thesis: verum

end;

suppose t > 1 ; :: thesis: ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2

then t >= 1 + 1 by NAT_1:13;
then t - 2 >= 2 - 2 by XREAL_1:9;
then (S-bound E) + 0 <= (S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ n)) * (t - 2)) by A5, XREAL_1:6;
hence ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2 by A17, A6, A9, XXREAL_0:2; :: thesis: verum

end;

end;

end;

((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `1 = ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `1 by A1, A2, A3, A4, A10, A13, A14, Th57;
hence (Gauge (E,n)) * ((Center (Gauge (E,n))),t) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) by A15, A18, A12, GOBOARD7:7; :: thesis: verum

end;

then ( (Gauge (E,n)) * ((Center (Gauge (E,n))),1) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) & (Gauge (E,n)) * ((Center (Gauge (E,n))),j) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) ) by A3;
hence LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) by TOPREAL1:6; :: thesis: verum