reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;
reserve V for set;

theorem Th82:
  for G2, V for G1 being addVertices of G2, V
  holds G1 == G2 iff V c= the_Vertices_of G2
proof
  let G2, V;
  let G1 be addVertices of G2, V;
  hereby
    assume G1 == G2;
    then the_Vertices_of G2 = the_Vertices_of G1 by GLIB_000:def 34
      .= the_Vertices_of G2 \/ V by Def10;
    hence V c= the_Vertices_of G2 by XBOOLE_1:11;
  end;
  assume V c= the_Vertices_of G2;
  then A1: V \/ the_Vertices_of G2 c=
    the_Vertices_of G2 \/ the_Vertices_of G2 by XBOOLE_1:9;
  the_Vertices_of G2 c= V \/ the_Vertices_of G2 by XBOOLE_1:7;
  then A2: the_Vertices_of G2 = V \/ the_Vertices_of G2 by A1, XBOOLE_0:def 10
    .= the_Vertices_of G1 by Def10;
  the_Edges_of G1 = the_Edges_of G2 & the_Source_of G1 = the_Source_of G2 &
    the_Target_of G1 = the_Target_of G2 by Def10;
  hence thesis by A2, GLIB_000:def 34;
end;
