let j, i be Element of 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 & j < width G ) and
A3: ( 1 <= i & 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 ) }
( 1 <= j + 1 & j + 1 <= width G ) by A2, NAT_1:13;
then ( (G * i,j) `2 = (G * 1,j) `2 & (G * i,(j + 1)) `2 = (G * 1,(j + 1)) `2 ) by A1, A2, A3, Th2;
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, Def2; :: thesis: verum