reserve f for non empty FinSequence of TOP-REAL 2,
  i,j,k,k1,k2,n,i1,i2,j1,j2 for Nat,
  r,s,r1,r2 for Real,
  p,q,p1,q1 for Point of TOP-REAL 2,
  G for Go-board;
reserve f for non constant standard special_circular_sequence;

theorem
  for p,q,p1,q1 be Point of TOP-REAL 2 st LSeg(p,q) is horizontal & LSeg
  (p1,q1) is horizontal & p`2 = p1`2 & p`1 <= p1`1 & p1`1 <= q1`1 & q1`1 <= q`1
  holds LSeg(p1,q1) c= LSeg(p,q)
proof
  let p,q,p1,q1 be Point of TOP-REAL 2;
  assume that
A1: LSeg(p,q) is horizontal and
A2: LSeg(p1,q1) is horizontal and
A3: p`2 = p1`2 and
A4: p`1 <= p1`1 and
A5: p1`1 <= q1`1 and
A6: q1`1 <= q`1;
A7: p`2 = q`2 by A1,SPPOL_1:15;
  let x be object;
  assume
A8: x in LSeg(p1,q1);
  then reconsider r=x as Point of TOP-REAL 2;
  p1`1 <= r`1 by A5,A8,TOPREAL1:3;
  then
A9: p`1 <= r`1 by A4,XXREAL_0:2;
  r`1 <= q1`1 by A5,A8,TOPREAL1:3;
  then
A10: r`1 <= q`1 by A6,XXREAL_0:2;
  p1`2 = r`2 by A2,A8,SPPOL_1:40;
  hence thesis by A3,A7,A9,A10,Th8;
end;
