let j be Nat; for G being Matrix of (TOP-REAL 2) st G is V3() & G is X_equal-in-line & 1 <= j & j <= width G holds
v_strip (G,(len G)) = { |[r,s]| where r, s is Real : (G * ((len G),j)) `1 <= r }
let G be Matrix of (TOP-REAL 2); ( G is V3() & G is X_equal-in-line & 1 <= j & j <= width G implies v_strip (G,(len G)) = { |[r,s]| where r, s is Real : (G * ((len G),j)) `1 <= r } )
assume that
A1:
( G is V3() & G is X_equal-in-line )
and
A2:
1 <= j
and
A3:
j <= width G
; v_strip (G,(len G)) = { |[r,s]| where r, s is Real : (G * ((len G),j)) `1 <= r }
len G <> 0
by A1, MATRIX_0:def 10;
then
1 <= len G
by NAT_1:14;
then
(G * ((len G),j)) `1 = (G * ((len G),1)) `1
by A1, A2, A3, Th2;
hence
v_strip (G,(len G)) = { |[r,s]| where r, s is Real : (G * ((len G),j)) `1 <= r }
by Def1; verum