
theorem Th66:
  for S being vertex-disjoint non empty Graph-membered set
  holds the set of all H.componentSet() where H is Element of S
    is mutually-disjoint
proof
  let S be vertex-disjoint non empty Graph-membered set;
  set M = the set of all H.componentSet() where H is Element of S;
  now
    let x,y be set;
    assume x in M;
    then consider H1 being Element of S such that
      A1: x = H1.componentSet();
    assume y in M;
    then consider H2 being Element of S such that
      A2: y = H2.componentSet();
    assume A3: x <> y;
    assume x meets y;
    then consider z being object such that
      A4: z in H1.componentSet() & z in H2.componentSet()
      by A1, A2, XBOOLE_0:3;
    reconsider z as set by TARSKI:1;
    A5: the_Vertices_of H1 misses the_Vertices_of H2 by A1, A2, A3, Def18;
    z c= the_Vertices_of H1 /\ the_Vertices_of H2 by A4, XBOOLE_1:19;
    then z c= {} by A5, XBOOLE_0:def 7;
    hence contradiction by A4;
  end;
  hence thesis by TAXONOM2:def 5;
end;
