reserve G for _Graph;
reserve V for non empty set, E for Relation of V;

theorem Th71:
  E is total reflexive iff createGraph(V,E) is loopfull
proof
  set G = createGraph(V,E);
  hereby
    assume E is total reflexive;
    then id V c= E by ROUGHS_1:3;
    then id the_Vertices_of G c= E;
    then id the_Vertices_of G c= VertexDomRel(G);
    hence createGraph(V,E) is loopfull by Lm2;
  end;
  assume G is loopfull;
  then id the_Vertices_of G c= VertexDomRel(G) by Lm2;
  then id the_Vertices_of G c= E;
  then id V c= E;
  hence thesis by ROUGHS_1:3;
end;
