let i be Nat; :: thesis: for G being Matrix of (TOP-REAL 2) st G is empty-yielding & 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); :: thesis: ( G is empty-yielding & 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 empty-yielding & G is Y_equal-in-column ) and
A2: 1 <= i and
A3: i <= len G ; :: thesis: 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; :: thesis: verum