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

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
A5: 1 <= t and
A6: 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)
A7: len (Gauge E,n) = width (Gauge E,n) by JORDAN8:def 1;
then A8: t <= len (Gauge E,n) by A4, A6, XXREAL_0:2;
A9: 0 < (N-bound E) - (S-bound E) by SPRECT_1:34, XREAL_1:52;
then A10: ((N-bound E) - (S-bound E)) / (2 |^ n) <= ((N-bound E) - (S-bound E)) / (2 |^ m) by A2, Lm7;
A11: 1 <= len (Gauge E,m) by GOBRD11:34;
then A12: ( ((Gauge E,m) * (Center (Gauge E,m)),1) `1 = ((Gauge E,n) * (Center (Gauge E,n)),t) `1 & ((Gauge E,n) * (Center (Gauge E,n)),t) `1 = ((Gauge E,n) * (Center (Gauge E,n)),j) `1 ) by A1, A2, A3, A4, A5, A7, A8, Th57;
A13: [(Center (Gauge E,n)),t] in Indices (Gauge E,n) by A5, A8, Lm4;
then A14: ((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:56 ;
((N-bound E) - (S-bound E)) / (2 |^ m) >= 0 by A9;
then A15: (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ m)) <= (S-bound E) - 0 by XREAL_1:15;
[(Center (Gauge E,m)),1] in Indices (Gauge E,m) by A11, Lm4;
then A16: ((Gauge E,m) * (Center (Gauge E,m)),1) `2 = (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ m)) by Lm11;
A17: ((N-bound E) - (S-bound E)) / (2 |^ n) >= 0 by A9;

A18: now

per cases ( t = 1 or t > 1 ) by A5, 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 A13, Lm11;
hence ((Gauge E,m) * (Center (Gauge E,m)),1) `2 <= ((Gauge E,n) * (Center (Gauge E,n)),t) `2 by A10, A16, XREAL_1:15; :: 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_1:11;
then (((N-bound E) - (S-bound E)) / (2 |^ n)) * (t - 2) >= 0 by A17;
then (S-bound E) + 0 <= (S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ n)) * (t - 2)) by XREAL_1:8;
hence ((Gauge E,m) * (Center (Gauge E,m)),1) `2 <= ((Gauge E,n) * (Center (Gauge E,n)),t) `2 by A14, A15, A16, XXREAL_0:2; :: thesis: verum

end;

end;

end;

( 1 <= Center (Gauge E,n) & Center (Gauge E,n) <= len (Gauge E,n) ) by Lm3;
then ((Gauge E,n) * (Center (Gauge E,n)),t) `2 <= ((Gauge E,n) * (Center (Gauge E,n)),j) `2 by A4, A5, A6, SPRECT_3:24;
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 A12, A18, GOBOARD7:8; :: 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 TOPREAL1:12; :: thesis: verum