let i, j be Nat; :: thesis: for G being Matrix of (TOP-REAL 2) st G is X_equal-in-line & 1 <= i & i < len G & 1 <= j & j <= width G holds
v_strip (G,i) = { |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 ) }
let G be Matrix of (TOP-REAL 2); :: thesis: ( G is X_equal-in-line & 1 <= i & i < len G & 1 <= j & j <= width G implies v_strip (G,i) = { |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 ) } )
assume that
A1: G is X_equal-in-line and
A2: 1 <= i and
A3: i < len G and
A4: 1 <= j and
A5: j <= width G ; :: thesis: v_strip (G,i) = { |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 ) }
A6: 1 <= i + 1 by A2, NAT_1:13;
A7: i + 1 <= len G by A3, NAT_1:13;
A8: (G * (i,j)) `1 = (G * (i,1)) `1 by A1, A2, A3, A4, A5, Th2;
(G * ((i + 1),j)) `1 = (G * ((i + 1),1)) `1 by A1, A4, A5, A6, A7, Th2;
hence v_strip (G,i) = { |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 ) } by A2, A3, A8, Def1; :: thesis: verum