
theorem Th39:
  for G1, G2 being _Graph, G3 being Component of G1 st G2 == G3
  holds G2 is Component of G1
proof
  let G1, G2 be _Graph, G3 be Component of G1;
  assume A1: G2 == G3;
  now
    thus G2 is connected by A1, GLIB_002:8;
    given G9 being connected Subgraph of G1 such that
      A2: G2 c< G9;
    A3: G2 c= G9 & G2 != G9 by A2, GLIB_000:def 36;
    then G2 is Subgraph of G9 by GLIB_000:def 35;
    then A4: G3 is Subgraph of G9 by A1, GLIB_000:92;
    G3 != G9 by A1, A3, GLIB_000:85;
    then G3 c< G9 by A4, GLIB_000:def 35, GLIB_000:def 36;
    hence contradiction by GLIB_002:def 7;
  end;
  hence thesis by A1, GLIB_000:92, GLIB_002:def 7;
end;
