
theorem
  for S1, S2 being Graph-membered set st S1 \/ S2 is vertex-disjoint
  holds S1 is vertex-disjoint & S2 is vertex-disjoint
proof
  let S1, S2 be Graph-membered set;
  assume A1: S1 \/ S2 is vertex-disjoint;
  hereby
    let G1, G2 be _Graph;
    assume G1 in S1 & G2 in S1 & G1 <> G2;
    then G1 in S1 \/ S2 & G2 in S1 \/ S2 & G1 <> G2 by XBOOLE_0:def 3;
    hence the_Vertices_of G1 misses the_Vertices_of G2 by A1;
  end;
  let G1, G2 be _Graph;
  assume G1 in S2 & G2 in S2 & G1 <> G2;
  then G1 in S1 \/ S2 & G2 in S1 \/ S2 & G1 <> G2 by XBOOLE_0:def 3;
  hence the_Vertices_of G1 misses the_Vertices_of G2 by A1;
end;
