let i, j be Nat; 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); ( 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
; 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; verum