
theorem
  for S1, S2 being Graph-membered set st S1,S2 are_Disomorphic holds
    (S1 is non-Dmulti implies S2 is non-Dmulti) &
    (S1 is Dsimple implies S2 is Dsimple)
proof
  let S1, S2 be Graph-membered set;
  assume S1,S2 are_Disomorphic;
  then consider f being one-to-one Function such that
    A1: dom f = S1 & rng f = S2 and
    A2: for G being _Graph st G in S1 holds f.G is G-Disomorphic _Graph;
  hereby
    assume A3: S1 is non-Dmulti;
    now
      let G2 be _Graph;
      assume G2 in S2;
      then consider G1 being object such that
        A4: G1 in dom f & f.G1 = G2 by A1, FUNCT_1:def 3;
      reconsider G1 as _Graph by A1, A4;
      G2 is G1-Disomorphic by A1, A2, A4;
      then consider G being PGraphMapping of G1, G2 such that
        A5: G is Disomorphism by GLIB_010:def 24;
      thus G2 is non-Dmulti by A1, A3, A4, A5, GLIB_010:90;
    end;
    hence S2 is non-Dmulti by GLIB_014:def 5;
  end;
  hereby
    assume A6: S1 is Dsimple;
    now
      let G2 be _Graph;
      assume G2 in S2;
      then consider G1 being object such that
        A7: G1 in dom f & f.G1 = G2 by A1, FUNCT_1:def 3;
      reconsider G1 as _Graph by A1, A7;
      G2 is G1-Disomorphic by A1, A2, A7;
      then consider G being PGraphMapping of G1, G2 such that
        A8: G is Disomorphism by GLIB_010:def 24;
      thus G2 is Dsimple by A1, A6, A7, A8, GLIB_010:90;
    end;
    hence S2 is Dsimple by GLIB_014:def 7;
  end;
end;
