
theorem Th49:
  for F1, F2 being Graph-yielding Function st F1, F2 are_Disomorphic holds
    (F1 is non-Dmulti implies F2 is non-Dmulti) &
    (F1 is Dsimple implies F2 is Dsimple)
proof
  let F1, F2 be Graph-yielding Function;
  assume F1,F2 are_Disomorphic;
  then consider p being one-to-one Function such that
    A1: dom p = dom F1 & rng p = dom F2 and
    A2: for x being object st x in dom F1 ex G1, G2 being _Graph
      st G1 = F1.x & G2 = F2.(p.x) & G2 is G1-Disomorphic;
  hereby
    assume A3: F1 is non-Dmulti;
    now
      let x be object;
      assume x in dom F2;
      then consider x0 being object such that
        A4: x0 in dom p & p.x0 = x by A1, FUNCT_1:def 3;
      consider G1, G2 being _Graph such that
        A5: G1 = F1.x0 & G2 = F2.(p.x0) & G2 is G1-Disomorphic by A1, A2, A4;
      take G2;
      thus F2.x = G2 by A4, A5;
      consider G9 being _Graph such that
        A6: F1.x0 = G9 & G9 is non-Dmulti by A1, A3, A4, GLIB_000:def 63;
      consider G being PGraphMapping of G1, G2 such that
        A7: G is Disomorphism by A5, GLIB_010:def 24;
      thus G2 is non-Dmulti by A5, A6, A7, GLIB_010:90;
    end;
    hence F2 is non-Dmulti by GLIB_000:def 63;
  end;
  hereby
    assume A8: F1 is Dsimple;
    now
      let x be object;
      assume x in dom F2;
      then consider x0 being object such that
        A9: x0 in dom p & p.x0 = x by A1, FUNCT_1:def 3;
      consider G1, G2 being _Graph such that
        A10: G1 = F1.x0 & G2 = F2.(p.x0) & G2 is G1-Disomorphic by A1, A2, A9;
      take G2;
      thus F2.x = G2 by A9, A10;
      consider G9 being _Graph such that
        A11: F1.x0 = G9 & G9 is Dsimple by A1, A8, A9, GLIB_000:def 65;
      consider G being PGraphMapping of G1, G2 such that
        A12: G is Disomorphism by A10, GLIB_010:def 24;
      thus G2 is Dsimple by A10, A11, A12, GLIB_010:90;
    end;
    hence F2 is Dsimple by GLIB_000:def 65;
  end;
end;
