let i, j be Nat; :: thesis: for G being Matrix of (TOP-REAL 2) st G is empty-yielding & G is X_equal-in-line & G is X_increasing-in-column & 1 <= i & i <= len G & 1 <= j & j < width G holds
LSeg ((G * (i,j)),(G * (i,(j + 1)))) c= v_strip (G,i)
let G be Matrix of (TOP-REAL 2); :: thesis: ( G is empty-yielding & G is X_equal-in-line & G is X_increasing-in-column & 1 <= i & i <= len G & 1 <= j & j < width G implies LSeg ((G * (i,j)),(G * (i,(j + 1)))) c= v_strip (G,i) )
assume that
A1: G is empty-yielding and
A2: G is X_equal-in-line and
A3: G is X_increasing-in-column and
A4: 1 <= i and
A5: i <= len G and
A6: 1 <= j and
A7: j < width G ; :: thesis: LSeg ((G * (i,j)),(G * (i,(j + 1)))) c= v_strip (G,i)
A8: 1 <= j + 1 by A6, NAT_1:13;
A9: j + 1 <= width G by A7, NAT_1:13;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) or x in v_strip (G,i) )
assume A10: x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) ; :: thesis: x in v_strip (G,i)
then reconsider p = x as Point of (TOP-REAL 2) ;
A11: p = |[(p `1),(p `2)]| by EUCLID:53;
A12: (G * (i,j)) `1 = (G * (i,1)) `1 by A2, A4, A5, A6, A7, Th2
.= (G * (i,(j + 1))) `1 by A2, A4, A5, A8, A9, Th2 ;

now :: thesis: x in v_strip (G,i)

per cases ( i = len G or i < len G ) by A5, XXREAL_0:1;

suppose A13: i = len G ; :: thesis: x in v_strip (G,i)

then (G * ((len G),j)) `1 <= p `1 by A10, A12, TOPREAL1:3;
then p in { |[r,s]| where r, s is Real : (G * ((len G),j)) `1 <= r } by A11;
hence x in v_strip (G,i) by A1, A2, A6, A7, A13, Th9; :: thesis: verum

end;

suppose A14: i < len G ; :: thesis: x in v_strip (G,i)

then A15: i + 1 <= len G by NAT_1:13;
A16: (G * (i,j)) `1 <= p `1 by A10, A12, TOPREAL1:3;
p `1 <= (G * (i,j)) `1 by A10, A12, TOPREAL1:3;
then A17: p `1 = (G * (i,j)) `1 by A16, XXREAL_0:1;
i < i + 1 by XREAL_1:29;
then p `1 < (G * ((i + 1),j)) `1 by A3, A4, A6, A7, A15, A17, Th3;
then p in { |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 ) } by A11, A16;
hence x in v_strip (G,i) by A2, A4, A6, A7, A14, Th8; :: thesis: verum

end;

end;

end;

hence x in v_strip (G,i) ; :: thesis: verum