reserve i,j,k,n,m for Nat;

theorem
  for p,q,r,s being Point of TOP-REAL 2 st LSeg(p,q) is horizontal &
  LSeg(r,s) is horizontal & LSeg(p,q) meets LSeg(r,s) holds p`2= r`2
proof
  let p,q,r,s be Point of TOP-REAL 2 such that
A1: LSeg(p,q) is horizontal and
A2: LSeg(r,s) is horizontal;
  assume LSeg(p,q) meets LSeg(r,s);
  then LSeg(p,q) /\ LSeg(r,s) <> {};
  then consider x being Point of TOP-REAL 2 such that
A3: x in LSeg(p,q) /\ LSeg(r,s) by SUBSET_1:4;
A4: x in LSeg(r,s) by A3,XBOOLE_0:def 4;
  x in LSeg(p,q) by A3,XBOOLE_0:def 4;
  hence p`2 = x`2 by A1,SPPOL_1:40
    .= r`2 by A2,A4,SPPOL_1:40;
end;
