let j be Nat; :: thesis: for G being Matrix of (TOP-REAL 2) st G is empty-yielding & G is X_equal-in-line & 1 <= j & j <= width G holds
v_strip (G,0) = { |[r,s]| where r, s is Real : r <= (G * (1,j)) `1 }
let G be Matrix of (TOP-REAL 2); :: thesis: ( G is empty-yielding & G is X_equal-in-line & 1 <= j & j <= width G implies v_strip (G,0) = { |[r,s]| where r, s is Real : r <= (G * (1,j)) `1 } )
assume that
A1: ( G is empty-yielding & G is X_equal-in-line ) and
A2: 1 <= j and
A3: j <= width G ; :: thesis: v_strip (G,0) = { |[r,s]| where r, s is Real : r <= (G * (1,j)) `1 }
set A = { |[r,s]| where r, s is Real : (G * (1,j)) `1 >= r } ;
A4: 0 <> len G by A1, MATRIX_0:def 10;
then A5: 0 < len G ;
1 <= len G by A4, NAT_1:14;
then (G * (1,j)) `1 = (G * (1,1)) `1 by A1, A2, A3, Th2;
then { |[r,s]| where r, s is Real : (G * (1,j)) `1 >= r } = { |[r,s]| where r, s is Real : (G * (1,(1 + 0))) `1 >= r } ;
hence v_strip (G,0) = { |[r,s]| where r, s is Real : r <= (G * (1,j)) `1 } by A5, Def1; :: thesis: verum