let i, n, j, m be Element of NAT ; :: thesis: for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge E,n) & 1 <= j & j <= len (Gauge E,m) & m <= n holds
((Gauge E,m) * j,1) `2 <= ((Gauge E,n) * i,1) `2
let E be compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ( 1 <= i & i <= len (Gauge E,n) & 1 <= j & j <= len (Gauge E,m) & m <= n implies ((Gauge E,m) * j,1) `2 <= ((Gauge E,n) * i,1) `2 )
set a = N-bound E;
set s = S-bound E;
set G = Gauge E,n;
set M = Gauge E,m;
assume that
A1: ( 1 <= i & i <= len (Gauge E,n) ) and
A2: ( 1 <= j & j <= len (Gauge E,m) ) and
A3: m <= n ; :: thesis: ((Gauge E,m) * j,1) `2 <= ((Gauge E,n) * i,1) `2
A4: len (Gauge E,m) = width (Gauge E,m) by JORDAN8:def 1;
1 <= len (Gauge E,m) by A2, XXREAL_0:2;
then [j,1] in Indices (Gauge E,m) by A2, A4, MATRIX_1:37;
then A5: ((Gauge E,m) * j,1) `2 = (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ m)) by Lm11;
A6: len (Gauge E,n) = width (Gauge E,n) by JORDAN8:def 1;
1 <= len (Gauge E,n) by A1, XXREAL_0:2;
then [i,1] in Indices (Gauge E,n) by A1, A6, MATRIX_1:37;
then ( 0 < (N-bound E) - (S-bound E) & ((Gauge E,n) * i,1) `2 = (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ n)) ) by Lm11, SPRECT_1:34, XREAL_1:52;
hence ((Gauge E,m) * j,1) `2 <= ((Gauge E,n) * i,1) `2 by A3, A5, Lm9; :: thesis: verum