let i, j be Element of NAT ; :: thesis: for G being Matrix of (TOP-REAL 2) st not 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: ( not 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: not G is empty-yielding and
A2: G is X_equal-in-line and
A3: G is X_increasing-in-column and
A4: ( 1 <= i & i <= len G ) and
A5: ( 1 <= j & j < width G ) ; :: thesis: LSeg (G * i,j),(G * i,(j + 1)) c= v_strip G,i
A6: ( 1 <= j + 1 & j + 1 <= width G ) by A5, NAT_1:13;
let x be set ; :: 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 A7: 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) ;
A8: p = |[(p `1 ),(p `2 )]| by EUCLID:57;
A9: (G * i,j) `1 = (G * i,1) `1 by A2, A4, A5, Th3
.= (G * i,(j + 1)) `1 by A2, A4, A6, Th3 ;

now

per cases ( i = len G or i < len G ) by A4, XXREAL_0:1;

suppose A10: i = len G ; :: thesis: x in v_strip G,i

then (G * (len G),j) `1 <= p `1 by A7, A9, TOPREAL1:9;
then p in { |[r,s]| where r, s is Real : (G * (len G),j) `1 <= r } by A8;
hence x in v_strip G,i by A1, A2, A5, A10, Th10; :: thesis: verum

end;

suppose A11: i < len G ; :: thesis: x in v_strip G,i

then A12: i + 1 <= len G by NAT_1:13;
A13: ( (G * i,j) `1 <= p `1 & p `1 <= (G * i,j) `1 ) by A7, A9, TOPREAL1:9;
then A14: p `1 = (G * i,j) `1 by XXREAL_0:1;
i < i + 1 by XREAL_1:31;
then p `1 < (G * (i + 1),j) `1 by A3, A4, A5, A12, A14, Th4;
then p in { |[r,s]| where r, s is Real : ( (G * i,j) `1 <= r & r <= (G * (i + 1),j) `1 ) } by A8, A13;
hence x in v_strip G,i by A2, A4, A5, A11, Th9; :: thesis: verum

end;

end;

end;

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