
theorem Th7:
  for S1,S2 being unsplit gate`1=arity non empty ManySortedSign st
  InputVertices S1 is without_pairs & InnerVertices S1 is Relation &
  InputVertices S2 is without_pairs & InnerVertices S2 is Relation holds
  InputVertices (S1+*S2) = (InputVertices S1) \/ (InputVertices S2)
proof
  let S1,S2 be unsplit gate`1=arity non empty ManySortedSign;
  assume InputVertices S1 is without_pairs & InnerVertices S1 is Relation &
  InputVertices S2 is without_pairs & InnerVertices S2 is Relation;
  then InputVertices (S1+*S2) = (InputVertices S1) \/ (InputVertices S2 \
  InnerVertices S1) & InnerVertices S1 misses InputVertices S2 by Th6;
  hence thesis by XBOOLE_1:83;
end;
