let i, j be Nat; :: thesis: for G being Matrix of (TOP-REAL 2) st G is Y_equal-in-column & G is Y_increasing-in-line & 1 <= i & i <= len G & 1 <= j & j + 1 <= width G holds
LSeg ((G * (i,j)),(G * (i,(j + 1)))) c= h_strip (G,j)
let G be Matrix of (TOP-REAL 2); :: thesis: ( G is Y_equal-in-column & G is Y_increasing-in-line & 1 <= i & i <= len G & 1 <= j & j + 1 <= width G implies LSeg ((G * (i,j)),(G * (i,(j + 1)))) c= h_strip (G,j) )
assume that
A1: G is Y_equal-in-column and
A2: G is Y_increasing-in-line and
A3: 1 <= i and
A4: i <= len G and
A5: 1 <= j and
A6: j + 1 <= width G ; :: thesis: LSeg ((G * (i,j)),(G * (i,(j + 1)))) c= h_strip (G,j)
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) or x in h_strip (G,j) )
assume A7: x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) ; :: thesis: x in h_strip (G,j)
then reconsider p = x as Point of (TOP-REAL 2) ;
A8: p = |[(p `1),(p `2)]| by EUCLID:53;
A9: j < width G by A6, NAT_1:13;
j < j + 1 by XREAL_1:29;
then A10: (G * (i,j)) `2 < (G * (i,(j + 1))) `2 by A2, A3, A4, A5, A6, Th4;
then A11: (G * (i,j)) `2 <= p `2 by A7, TOPREAL1:4;
p `2 <= (G * (i,(j + 1))) `2 by A7, A10, TOPREAL1:4;
then p in { |[r,s]| where r, s is Real : ( (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 ) } by A8, A11;
hence x in h_strip (G,j) by A1, A3, A4, A5, A9, Th5; :: thesis: verum