
theorem Th68:
  for S being vertex-disjoint GraphUnionSet, G being GraphUnion of S
  holds card S c= G.numComponents()
proof
  let S be vertex-disjoint GraphUnionSet, G be GraphUnion of S;
  set M = the set of all H.componentSet() where H is Element of S;
  now
    thus M is mutually-disjoint by Th66;
    thus card S c= card M by Lm2;
    let Y be set;
    assume Y in M;
    then consider H being Element of S such that
      A1: Y = H.componentSet();
    thus 1 c= card Y by A1, Lm3;
  end;
  then (card S)*`1 c= card union M by GLIBPRE1:15;
  then card S c= card union M by CARD_2:21;
  then card S c= card G.componentSet() by Th65;
  hence thesis by GLIB_002:def 9;
end;
