
theorem
  for G1, G2 being _Graph holds {G1, G2} is vertex-disjoint
    iff G1 = G2 or the_Vertices_of G1 misses the_Vertices_of G2
proof
  let G1, G2 be _Graph;
  hereby
    assume A1: {G1, G2} is vertex-disjoint;
    assume A2: G1 <> G2;
    G1 in {G1, G2} & G2 in {G1, G2} by TARSKI:def 2;
    hence the_Vertices_of G1 misses the_Vertices_of G2 by A1, A2;
  end;
  assume G1 = G2 or the_Vertices_of G1 misses the_Vertices_of G2;
  then per cases;
  suppose G1 = G2;
    then {G1, G2} = {G1} by ENUMSET1:29;
    hence thesis;
  end;
  suppose A3: the_Vertices_of G1 misses the_Vertices_of G2;
    let G3, G4 be _Graph;
    assume G3 in {G1,G2} & G4 in {G1,G2} & G3 <> G4;
    then (G3=G1 or G3=G2) & (G4=G1 or G4=G2) & G3 <> G4 by TARSKI:def 2;
    hence thesis by A3;
  end;
end;
