reserve p,q for Point of TOP-REAL 2,
  i,i1,i2,j,j1,j2,k for Nat,
  r,s for Real,
  G for Matrix of TOP-REAL 2;
reserve f for standard special_circular_sequence;

theorem Th17:
  G is Y_equal-in-column 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)
proof
  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;
  let x be object;
  assume
A7: x in LSeg(G*(i,j),G*(i,j+1));
  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]| : G*(i,j)`2 <= s & s <= G*(i,j+1)`2 } by A8,A11;
  hence thesis by A1,A3,A4,A5,A9,Th5;
end;
