let i, j be Nat; :: thesis: for G being Matrix of (TOP-REAL 2) st G is Y_equal-in-column & 1 <= j & j < width G & 1 <= i & i <= len G holds
h_strip (G,j) = { |[r,s]| where r, s is Real : ( (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 ) }
let G be Matrix of (TOP-REAL 2); :: thesis: ( G is Y_equal-in-column & 1 <= j & j < width G & 1 <= i & i <= len G implies h_strip (G,j) = { |[r,s]| where r, s is Real : ( (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 ) } )
assume that
A1: G is Y_equal-in-column and
A2: 1 <= j and
A3: j < width G and
A4: 1 <= i and
A5: i <= len G ; :: thesis: h_strip (G,j) = { |[r,s]| where r, s is Real : ( (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 ) }
A6: 1 <= j + 1 by A2, NAT_1:13;
A7: j + 1 <= width G by A3, NAT_1:13;
A8: (G * (i,j)) `2 = (G * (1,j)) `2 by A1, A2, A3, A4, A5, Th1;
(G * (i,(j + 1))) `2 = (G * (1,(j + 1))) `2 by A1, A4, A5, A6, A7, Th1;
hence h_strip (G,j) = { |[r,s]| where r, s is Real : ( (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 ) } by A2, A3, A8, Def2; :: thesis: verum