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

theorem Th85:
  for G2, V for G1 being addVertices of G2, V
  holds G1.edgesBetween(the_Vertices_of G2) = the_Edges_of G1
proof
  let G2, V;
  let G1 be addVertices of G2, V;
  set E1 = the_Edges_of G1;
  set V2 = the_Vertices_of G2;
  for e being object holds e in E1 iff
    e in G1.edgesInto(V2) /\ G1.edgesOutOf(V2)
  proof
    let e be object;
    hereby
      assume A1: e in the_Edges_of G1;
      then A2: e in the_Edges_of G2 by Def10;
      then A3: (the_Source_of G2).e in V2 & (the_Target_of G2).e in V2
        by FUNCT_2:5;
      reconsider e1=e as set by TARSKI:1;
      (the_Source_of G1).e1 in V2 & (the_Target_of G1).e1 in V2
        by A2, A3, Def9;
      then e1 in G1.edgesInto(V2) & e1 in G1.edgesOutOf(V2)
        by A1, GLIB_000:def 26, GLIB_000:def 27;
      hence e in G1.edgesInto(V2) /\ G1.edgesOutOf(V2) by XBOOLE_0:def 4;
    end;
    thus thesis;
  end;
  then E1 = G1.edgesInto(V2) /\ G1.edgesOutOf(V2) by TARSKI:2;
  hence thesis by GLIB_000:def 29;
end;
