reserve i, i1, i2, j, j1, j2, k, m, n, t for Nat,
  D for non empty Subset of TOP-REAL 2,
  E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p, q, x for Point of TOP-REAL 2,
  r, s for Real;

theorem
  for A,B being Subset of TOP-REAL 2 st A misses B & A c= Vertical_Line
  s & B c= Vertical_Line s holds proj2.:A misses proj2.:B
proof
  let A,B being Subset of TOP-REAL 2 such that
A1: A misses B and
A2: A c= Vertical_Line s and
A3: B c= Vertical_Line s;
  assume proj2.:A meets proj2.:B;
  then consider e being object such that
A4: e in proj2.:A and
A5: e in proj2.:B by XBOOLE_0:3;
  reconsider e as Real by A4;
  consider d being Point of TOP-REAL 2 such that
A6: d in B and
A7: e = proj2.d by A5,FUNCT_2:65;
A8: d`1 = s by A3,A6,JORDAN6:31;
  consider c being Point of TOP-REAL 2 such that
A9: c in A and
A10: e = proj2.c by A4,FUNCT_2:65;
  c`1 = s by A2,A9,JORDAN6:31;
  then c = |[d`1,c`2]| by A8,EUCLID:53
    .= |[d`1,e]| by A10,PSCOMP_1:def 6
    .= |[d`1,d`2]| by A7,PSCOMP_1:def 6
    .= d by EUCLID:53;
  hence contradiction by A1,A9,A6,XBOOLE_0:3;
end;
