let i, j be Element of 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 & i < len G ) and
A3: ( 1 <= j & 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 ) }
( 1 <= i + 1 & i + 1 <= len G ) by A2, NAT_1:13;
then ( (G * i,j) `1 = (G * i,1) `1 & (G * (i + 1),j) `1 = (G * (i + 1),1) `1 ) by A1, A2, A3, Th3;
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, Def1; :: thesis: verum