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 Th15:
  G is non empty-yielding Y_equal-in-column Y_increasing-in-line &
  j < width G & 1 <= i & i < len G
  implies LSeg(G*(i,j+1),G*(i+1,j+1)) c= h_strip(G,j)
proof
  assume that
A1: G is non empty-yielding and
A2: G is Y_equal-in-column and
A3: G is Y_increasing-in-line and
A4: j < width G and
A5: 1 <= i and
A6: i < len G;
A7: 1 <= i+1 by A5,NAT_1:13;
A8: i+1 <= len G by A6,NAT_1:13;
  let x be object;
  assume
A9: x in LSeg(G*(i,j+1),G*(i+1,j+1));
  then reconsider p = x as Point of TOP-REAL 2;
A10: p = |[p`1, p`2]| by EUCLID:53;
A11: 1 <= j+1 by NAT_1:11;
A12: j+1 <= width G by A4,NAT_1:13;
  then
A13: G*(i,j+1)`2 = G*(1,j+1)`2 by A2,A5,A6,A11,Th1
    .= G*(i+1,j+1)`2 by A2,A7,A8,A11,A12,Th1;
  now per cases;
    suppose
A14:  j = 0;
      then p`2 <= G*(i,1)`2 by A9,A13,TOPREAL1:4;
      then p in { |[r,s]| : s <= G*(i,1)`2 } by A10;
      hence thesis by A1,A2,A5,A6,A14,Th7;
    end;
    suppose j > 0;
      then
A15:  0+1 <= j by NAT_1:13;
A16:  G*(i,j+1)`2 <= p`2 by A9,A13,TOPREAL1:4;
A17:  p`2 <= G*(i,j+1)`2 by A9,A13,TOPREAL1:4;
      then
A18:  p`2 = G*(i,j+1)`2 by A16,XXREAL_0:1;
      j < j+1 by XREAL_1:29;
      then G*(i,j)`2 < p`2 by A3,A5,A6,A12,A15,A18,Th4;
      then p in { |[r,s]| : G*(i,j)`2 <= s & s <= G*(i,j+1)`2 } by A10,A17;
      hence thesis by A2,A4,A5,A6,A15,Th5;
    end;
  end;
  hence thesis;
end;
