
theorem Th69:
  for S being vertex-disjoint GraphUnionSet, G being GraphUnion of S
  st S is connected holds card S = G.numComponents()
proof
  let S be vertex-disjoint GraphUnionSet, G be GraphUnion of S;
  assume A1: S is connected;
  thus card S
     = card the set of all H.componentSet() where H is Element of S by Lm2
    .= card SmallestPartition the_Vertices_of S by A1, Th67
    .= card the_Vertices_of S by TOPGEN_2:12
    .= card union SmallestPartition the_Vertices_of S by EQREL_1:def 4
    .= card union the set of all H.componentSet() where H is Element of S
      by A1, Th67
    .= card G.componentSet() by Th65
    .= G.numComponents() by GLIB_002:def 9;
end;
