
theorem
  for G1, G2 being _Graph holds <* G1, G2 *> is vertex-disjoint iff
    the_Vertices_of G1 misses the_Vertices_of G2
proof
  let G1, G2 be _Graph;
  set F = <* G1, G2 *>;
  hereby
    assume A1: F is vertex-disjoint;
    1 in {1,2} & 2 in {1,2} by TARSKI:def 2;
    then reconsider x1 = 1, x2 = 2 as Element of dom F by FINSEQ_1:92;
    the_Vertices_of (F.x1) misses the_Vertices_of (F.x2) by A1;
    then the_Vertices_of G1 misses the_Vertices_of (F.x2);
    hence the_Vertices_of G1 misses the_Vertices_of G2;
  end;
  assume A2: the_Vertices_of G1 misses the_Vertices_of G2;
  let x1, x2 be Element of dom F;
  assume A3: x1 <> x2;
  x1 in dom F & x2 in dom F;
  then x1 in {1,2} & x2 in {1,2} by FINSEQ_1:92;
  then (x1 = 1 or x1 = 2) & (x2 = 1 or x2 = 2) by TARSKI:def 2;
  then per cases by A3;
  suppose x1 = 1 & x2 = 2;
    then F.x1 = G1 & F.x2 = G2;
    hence thesis by A2;
  end;
  suppose x1 = 2 & x2 = 1;
    then F.x2 = G1 & F.x1 = G2;
    hence thesis by A2;
  end;
end;
