let i, j, m, n be 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_0:30;
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_0:30;
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_1:32, XREAL_1:50;
hence ((Gauge (E,m)) * (j,1)) `2 <= ((Gauge (E,n)) * (i,1)) `2 by A3, A5, Lm9; :: thesis: verum