let j be Element of NAT ; :: thesis: for G being Matrix of (TOP-REAL 2) st not 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: ( not 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: ( not G is empty-yielding & G is X_equal-in-line ) and
A2: ( 1 <= j & 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 } ;
A3: 0 <> len G by A1, GOBOARD1:def 5;
then A4: 0 < len G by NAT_1:3;
1 <= len G by A3, NAT_1:14;
then (G * 1,j) `1 = (G * 1,1) `1 by A1, A2, Th3;
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 A4, Def1; :: thesis: verum