reserve G for _Graph;

theorem Th40:
  G is loopfull iff VertexAdjSymRel(G) is total reflexive
proof
  hereby
    assume G is loopfull;
    then id the_Vertices_of G c= VertexAdjSymRel(G) by Lm4;
    hence VertexAdjSymRel(G) is total reflexive by ROUGHS_1:3;
  end;
  assume VertexAdjSymRel(G) is total reflexive;
  then id the_Vertices_of G c= VertexAdjSymRel(G) by ROUGHS_1:3;
  hence G is loopfull by Lm4;
end;
