let k, n, i, j be Element of NAT ; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= k & k + 1 <= len (Cage C,n) & [i,j] in Indices (Gauge C,n) & [(i + 1),j] in Indices (Gauge C,n) & (Cage C,n) /. k = (Gauge C,n) * i,j & (Cage C,n) /. (k + 1) = (Gauge C,n) * (i + 1),j holds
j > 1
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ( 1 <= k & k + 1 <= len (Cage C,n) & [i,j] in Indices (Gauge C,n) & [(i + 1),j] in Indices (Gauge C,n) & (Cage C,n) /. k = (Gauge C,n) * i,j & (Cage C,n) /. (k + 1) = (Gauge C,n) * (i + 1),j implies j > 1 )
set f = Cage C,n;
set G = Gauge C,n;
assume that
A1: ( 1 <= k & k + 1 <= len (Cage C,n) ) and
A2: [i,j] in Indices (Gauge C,n) and
A3: ( [(i + 1),j] in Indices (Gauge C,n) & (Cage C,n) /. k = (Gauge C,n) * i,j & (Cage C,n) /. (k + 1) = (Gauge C,n) * (i + 1),j ) ; :: thesis: j > 1
assume A4: j <= 1 ; :: thesis: contradiction
1 <= j by A2, MATRIX_1:39;
then j = 1 by A4, XXREAL_0:1;
then A5: j -' 1 = 0 by XREAL_1:234;
A6: i <= len (Gauge C,n) by A2, MATRIX_1:39;
Cage C,n is_sequence_on Gauge C,n by JORDAN9:def 1;
then right_cell (Cage C,n),k,(Gauge C,n) = cell (Gauge C,n),i,(j -' 1) by A1, A2, A3, GOBRD13:25;
hence contradiction by A1, A5, A6, JORDAN8:20, JORDAN9:33; :: thesis: verum