let i, j be Nat; for G being Matrix of (TOP-REAL 2) st G is X_equal-in-line & G is X_increasing-in-column & 1 <= j & j <= width G & 1 <= i & i + 1 <= len G holds
LSeg ((G * (i,j)),(G * ((i + 1),j))) c= v_strip (G,i)
let G be Matrix of (TOP-REAL 2); ( G is X_equal-in-line & G is X_increasing-in-column & 1 <= j & j <= width G & 1 <= i & i + 1 <= len G implies LSeg ((G * (i,j)),(G * ((i + 1),j))) c= v_strip (G,i) )
assume that
A1:
G is X_equal-in-line
and
A2:
G is X_increasing-in-column
and
A3:
1 <= j
and
A4:
j <= width G
and
A5:
1 <= i
and
A6:
i + 1 <= len G
; LSeg ((G * (i,j)),(G * ((i + 1),j))) c= v_strip (G,i)
let x be object ; TARSKI:def 3 ( not x in LSeg ((G * (i,j)),(G * ((i + 1),j))) or x in v_strip (G,i) )
assume A7:
x in LSeg ((G * (i,j)),(G * ((i + 1),j)))
; x in v_strip (G,i)
then reconsider p = x as Point of (TOP-REAL 2) ;
A8:
p = |[(p `1),(p `2)]|
by EUCLID:53;
A9:
i < len G
by A6, NAT_1:13;
i < i + 1
by XREAL_1:29;
then A10:
(G * (i,j)) `1 < (G * ((i + 1),j)) `1
by A2, A3, A4, A5, A6, Th3;
then A11:
(G * (i,j)) `1 <= p `1
by A7, TOPREAL1:3;
p `1 <= (G * ((i + 1),j)) `1
by A7, A10, TOPREAL1:3;
then
p in { |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 ) }
by A8, A11;
hence
x in v_strip (G,i)
by A1, A3, A4, A5, A9, Th8; verum