reserve n for Nat;

theorem
  for A,B be Subset of TOP-REAL 2 for s be Real st A misses B & A
  c= Horizontal_Line s & B c= Horizontal_Line s holds proj1.:A misses proj1.:B
proof
  let A,B be Subset of TOP-REAL 2;
  let s be Real such that
A1: A misses B and
A2: A c= Horizontal_Line s and
A3: B c= Horizontal_Line s;
  assume proj1.:A meets proj1.:B;
  then consider e be object such that
A4: e in proj1.:A and
A5: e in proj1.:B by XBOOLE_0:3;
  reconsider e as Real by A4;
  consider d be Point of TOP-REAL 2 such that
A6: d in B and
A7: e = proj1.d by A5,FUNCT_2:65;
A8: d`2 = s by A3,A6,JORDAN6:32;
  consider c being Point of TOP-REAL 2 such that
A9: c in A and
A10: e = proj1.c by A4,FUNCT_2:65;
  c`2 = s by A2,A9,JORDAN6:32;
  then c = |[c`1,d`2]| by A8,EUCLID:53
    .= |[e,d`2]| by A10,PSCOMP_1:def 5
    .= |[d`1,d`2]| by A7,PSCOMP_1:def 5
    .= d by EUCLID:53;
  hence contradiction by A1,A9,A6,XBOOLE_0:3;
end;
