let i be Nat; for G being Matrix of (TOP-REAL 2) st G is V3() & G is Y_equal-in-column & 1 <= i & i <= len G holds
h_strip (G,0) = { |[r,s]| where r, s is Real : s <= (G * (i,1)) `2 }
let G be Matrix of (TOP-REAL 2); ( G is V3() & G is Y_equal-in-column & 1 <= i & i <= len G implies h_strip (G,0) = { |[r,s]| where r, s is Real : s <= (G * (i,1)) `2 } )
assume that
A1:
( G is V3() & G is Y_equal-in-column )
and
A2:
1 <= i
and
A3:
i <= len G
; h_strip (G,0) = { |[r,s]| where r, s is Real : s <= (G * (i,1)) `2 }
set A = { |[r,s]| where r, s is Real : (G * (i,1)) `2 >= s } ;
A4:
0 <> width G
by A1, MATRIX_0:def 10;
then A5:
0 < width G
;
1 <= width G
by A4, NAT_1:14;
then
(G * (i,1)) `2 = (G * (1,1)) `2
by A1, A2, A3, Th1;
then
{ |[r,s]| where r, s is Real : (G * (i,1)) `2 >= s } = { |[r,s]| where r, s is Real : (G * (1,(1 + 0))) `2 >= s }
;
hence
h_strip (G,0) = { |[r,s]| where r, s is Real : s <= (G * (i,1)) `2 }
by A5, Def2; verum