
theorem Th9:
  for G1, G2 being non-Dmulti _Graph, f being PVertexMapping of G1, G2
  st f is total one-to-one directed Dcontinuous for v,w being Vertex of G1
  holds card G1.edgesDBetween({v},{w}) = card G2.edgesDBetween({f.v},{f.w}) &
    card G1.edgesDBetween({w},{v}) = card G2.edgesDBetween({f.w},{f.v})
proof
  let G1, G2 be non-Dmulti _Graph, f be PVertexMapping of G1, G2;
  assume A1: f is total one-to-one directed Dcontinuous;
  let v,w be Vertex of G1;
  hereby
    per cases;
    suppose A2: G1.edgesDBetween({v},{w}) = {};
      G2.edgesDBetween({f.v},{f.w}) = {}
      proof
        assume G2.edgesDBetween({f.v},{f.w}) <> {};
        then consider e2 being object such that
          A3: G2.edgesDBetween({f.v},{f.w}) = {e2}
          by ZFMISC_1:131;
        set v1 = (the_Source_of G2).e2, v2 = (the_Target_of G2).e2;
        e2 in G2.edgesDBetween({f.v},{f.w}) by A3, TARSKI:def 1;
        then e2 DSJoins {f.v},{f.w},G2 by GLIB_000:def 31;
        then A4: e2 in the_Edges_of G2 & v1 in {f.v} & v2 in {f.w}
          by GLIB_000:def 16;
        then A5: v1 = f.v & v2 = f.w by TARSKI:def 1;
        v in the_Vertices_of G1 & w in the_Vertices_of G1 & f is total by A1;
        then v in dom f & w in dom f by FUNCT_2:def 1;
        then consider e1 being object such that
          A6: e1 DJoins v,w,G1 by A1, A4, A5, GLIB_000:def 14;
        v in {v} & w in {w} by TARSKI:def 1;
        then e1 DSJoins {v},{w},G1 by A6, GLIB_000:126;
        hence contradiction by A2, GLIB_000:def 31;
      end;
      hence card G1.edgesDBetween({v},{w})
        = card G2.edgesDBetween({f.v},{f.w}) by A2;
    end;
    suppose G1.edgesDBetween({v},{w}) <> {};
      then consider e1 being object such that
        A7: G1.edgesDBetween({v},{w}) = {e1} by ZFMISC_1:131;
      set v1 = (the_Source_of G1).e1, v2 = (the_Target_of G1).e1;
      e1 in G1.edgesDBetween({v},{w}) by A7, TARSKI:def 1;
      then e1 DSJoins {v},{w},G1 by GLIB_000:def 31;
      then A8: e1 in the_Edges_of G1 & v1 in {v} & v2 in {w}
        by GLIB_000:def 16;
      then v1 = v & v2 = w by TARSKI:def 1;
      then A9: e1 DJoins v,w,G1 by A8, GLIB_000:def 14;
      v in the_Vertices_of G1 & w in the_Vertices_of G1 & f is total by A1;
      then v in dom f & w in dom f by FUNCT_2:def 1;
      then consider e2 being object such that
        A10: e2 DJoins f.v,f.w,G2 by A1, A9;
      f.v in {f.v} & f.w in {f.w} by TARSKI:def 1;
      then e2 DSJoins {f.v},{f.w},G2 by A10, GLIB_000:126;
      then e2 in G2.edgesDBetween({f.v},{f.w}) by GLIB_000:def 31;
      then consider e being object such that
        A11: G2.edgesDBetween({f.v},{f.w}) = {e} by ZFMISC_1:131;
      card G1.edgesDBetween({v},{w}) = 1 &
        card G2.edgesDBetween({f.v},{f.w}) = 1 by A7, A11, CARD_1:30;
      hence card G1.edgesDBetween({v},{w})
        = card G2.edgesDBetween({f.v},{f.w});
    end;
  end;
  per cases;
  suppose A12: G1.edgesDBetween({w},{v}) = {};
    G2.edgesDBetween({f.w},{f.v}) = {}
    proof
      assume G2.edgesDBetween({f.w},{f.v}) <> {};
      then consider e2 being object such that
        A13: G2.edgesDBetween({f.w},{f.v}) = {e2} by ZFMISC_1:131;
      set v1 = (the_Source_of G2).e2, v2 = (the_Target_of G2).e2;
      e2 in G2.edgesDBetween({f.w},{f.v}) by A13, TARSKI:def 1;
      then e2 DSJoins {f.w},{f.v},G2 by GLIB_000:def 31;
      then A14: e2 in the_Edges_of G2 & v1 in {f.w} & v2 in {f.v}
        by GLIB_000:def 16;
      then A15: v1 = f.w & v2 = f.v by TARSKI:def 1;
      v in the_Vertices_of G1 & w in the_Vertices_of G1 & f is total by A1;
      then v in dom f & w in dom f by FUNCT_2:def 1;
      then consider e1 being object such that
        A16: e1 DJoins w,v,G1 by A1, A14, A15, GLIB_000:def 14;
      v in {v} & w in {w} by TARSKI:def 1;
      then e1 DSJoins {w},{v},G1 by A16, GLIB_000:126;
      hence contradiction by A12, GLIB_000:def 31;
    end;
    hence card G1.edgesDBetween({w},{v})
      = card G2.edgesDBetween({f.w},{f.v}) by A12;
  end;
  suppose G1.edgesDBetween({w},{v}) <> {};
    then consider e1 being object such that
      A17: G1.edgesDBetween({w},{v}) = {e1} by ZFMISC_1:131;
    set v1 = (the_Source_of G1).e1, v2 = (the_Target_of G1).e1;
    e1 in G1.edgesDBetween({w},{v}) by A17, TARSKI:def 1;
    then e1 DSJoins {w},{v},G1 by GLIB_000:def 31;
    then A18: e1 in the_Edges_of G1 & v1 in {w} & v2 in {v}
      by GLIB_000:def 16;
    then v1 = w & v2 = v by TARSKI:def 1;
    then A19: e1 DJoins w,v,G1 by A18, GLIB_000:def 14;
    v in the_Vertices_of G1 & w in the_Vertices_of G1 & f is total by A1;
    then v in dom f & w in dom f by FUNCT_2:def 1;
    then consider e2 being object such that
      A20: e2 DJoins f.w,f.v,G2 by A1, A19;
    f.v in {f.v} & f.w in {f.w} by TARSKI:def 1;
    then e2 DSJoins {f.w},{f.v},G2 by A20, GLIB_000:126;
    then e2 in G2.edgesDBetween({f.w},{f.v}) by GLIB_000:def 31;
    then consider e being object such that
      A21: G2.edgesDBetween({f.w},{f.v}) = {e} by ZFMISC_1:131;
    card G1.edgesDBetween({w},{v}) = 1 &
      card G2.edgesDBetween({f.w},{f.v}) = 1 by A17, A21, CARD_1:30;
    hence card G1.edgesDBetween({w},{v})
      = card G2.edgesDBetween({f.w},{f.v});
  end;
end;
