
theorem Th8:
  for S1,S2 being non empty ManySortedSign st S1 tolerates S2 &
  InputVertices S1 is without_pairs & InputVertices S2 is without_pairs holds
  InputVertices (S1+*S2) is without_pairs
proof
  let S1, S2 be non empty ManySortedSign;
  assume S1 tolerates S2;
  then
A1: InputVertices (S1+*S2) c= (InputVertices S1) \/ InputVertices S2 by
CIRCCOMB:11;
  assume
A2: for x being pair object holds not x in InputVertices S1;
  assume
A3: for x being pair object holds not x in InputVertices S2;
  let x be pair object;
  assume x in InputVertices (S1+*S2);
  then x in InputVertices S1 or x in InputVertices S2 by A1,XBOOLE_0:def 3;
  hence contradiction by A2,A3;
end;
