
theorem Th38:
  for S being GraphMeetSet, G being GraphMeet of S, G9 being _Graph
  holds G9 is GraphMeet of S iff G == G9
proof
  let S be GraphMeetSet, G be GraphMeet of S, G9 be _Graph;
  A1: the_Vertices_of G = meet the_Vertices_of S &
    the_Edges_of G = meet the_Edges_of S &
    the_Source_of G = meet the_Source_of S &
    the_Target_of G = meet the_Target_of S by Def29;
  hereby
    assume G9 is GraphMeet of S;
    then the_Vertices_of G9 = meet the_Vertices_of S &
      the_Edges_of G9 = meet the_Edges_of S &
      the_Source_of G9 = meet the_Source_of S &
      the_Target_of G9 = meet the_Target_of S by Def29;
    hence G == G9 by A1, GLIB_000:def 34;
  end;
  assume G == G9;
  then the_Vertices_of G = the_Vertices_of G9 &
    the_Edges_of G = the_Edges_of G9 &
    the_Source_of G = the_Source_of G9 &
    the_Target_of G = the_Target_of G9 by GLIB_000:def 34;
  hence thesis by A1, Def29;
end;
