
theorem Th43:
  for G1, G2, G9 being _Graph, G being GraphMeet of G1, G2
  holds G9 is GraphMeet of G1, G2 iff G == G9
proof
  let G1, G2, G9 be _Graph, G be GraphMeet of G1, G2;
  hereby
    assume A1: G9 is GraphMeet of G1, G2;
    per cases;
    suppose A2: G1 tolerates G2 & the_Vertices_of G1 meets the_Vertices_of G2;
      then consider S being GraphMeetSet such that
        A3: S = {G1,G2} & G is GraphMeet of S by Def30;
      consider S9 being GraphMeetSet such that
        A4: S9 = {G1,G2} & G9 is GraphMeet of S9 by A1, A2, Def30;
      thus G == G9 by A3, A4, Th38;
    end;
    suppose not(G1 tolerates G2 & the_Vertices_of G1 meets the_Vertices_of G2);
      then G == G1 & G9 == G1 by A1, Def30;
      hence G == G9 by GLIB_000:85;
    end;
  end;
  assume A5: G == G9;
  per cases;
  suppose A6: G1 tolerates G2 & the_Vertices_of G1 meets the_Vertices_of G2;
    then consider S being GraphMeetSet such that
      A7: S = {G1,G2} & G is GraphMeet of S by Def30;
    A8: G9 is Subgraph of G1 by A5, GLIB_000:92;
    G9 is GraphMeet of S by A5, A7, Th38;
    hence thesis by A6, A7, A8, Def30;
  end;
  suppose A9: not(G1 tolerates G2
      & the_Vertices_of G1 meets the_Vertices_of G2);
    then G == G1 by Def30;
    then A10: G9 == G1 by A5, GLIB_000:85;
    then G9 is Subgraph of G1 by GLIB_006:58;
    hence thesis by A9, A10, Def30;
  end;
end;
