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

theorem
  E is symmetric iff VertexAdjSymRel(createGraph(V,E)) = E
proof
  set G = createGraph(V,E);
  hereby
    assume E is symmetric;
    then A1: E = E~ by RELAT_2:13;
    thus VertexAdjSymRel(G) = E \/ ((VertexDomRel(G))~)
      .= E \/ E~
      .= E by A1;
  end;
  assume VertexAdjSymRel(G) = E;
  then E = E \/ ((VertexDomRel(G))~)
    .= E \/ E~;
  then A2: E~ c= E by XBOOLE_1:7;
  then E~~ c= E~ by SYSREL:11;
  hence thesis by A2, XBOOLE_0:def 10, RELAT_2:13;
end;
