
theorem Th37:
  for S being GraphMeetSet, G being GraphMeet of S, H being Element of S
  holds H is Supergraph of G
proof
  let S be GraphMeetSet, G be GraphMeet of S, H be Element of S;
  the_Vertices_of H in the_Vertices_of S;
  then meet the_Vertices_of S c= the_Vertices_of H by SETFAM_1:3;
  then A1: the_Vertices_of G c= the_Vertices_of H by Def29;
  the_Source_of H in the_Source_of S;
  then meet the_Source_of S c= the_Source_of H by SETFAM_1:3;
  then A2: the_Source_of G c= the_Source_of H by Def29;
  the_Target_of H in the_Target_of S;
  then meet the_Target_of S c= the_Target_of H by SETFAM_1:3;
  then the_Target_of G c= the_Target_of H by Def29;
  hence H is Supergraph of G by A1, A2, GLIB_006:63;
end;
