
theorem Th8:
  for G1, G2 being non-multi _Graph, f being PVertexMapping of G1, G2
  st f is total one-to-one continuous for v,w being Vertex of G1
  holds card G1.edgesBetween({v},{w}) = card G2.edgesBetween({f.v},{f.w})
proof
  let G1, G2 be non-multi _Graph, f be PVertexMapping of G1, G2;
  assume A1: f is total one-to-one continuous;
  let v,w be Vertex of G1;
  per cases;
  suppose A2: G1.edgesBetween({v},{w}) = {};
    G2.edgesBetween({f.v},{f.w}) = {}
    proof
      assume G2.edgesBetween({f.v},{f.w}) <> {};
      then consider e2 being object such that
        A3: G2.edgesBetween({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.edgesBetween({f.v},{f.w}) by A3, TARSKI:def 1;
      then e2 SJoins {f.v},{f.w},G2 by GLIB_000:def 30;
      then A4: e2 in the_Edges_of G2 & (v1 in {f.v} & v2 in {f.w} or
        v1 in {f.w} & v2 in {f.v}) by GLIB_000:def 15;
      then v1 = f.v & v2 = f.w or v1 = f.w & v2 = f.v by TARSKI:def 1;
      then A5: e2 Joins f.v, f.w, G2 by A4, GLIB_000:def 13;
      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 Joins v,w,G1 by A1, A5, Th2;
      v in {v} & w in {w} by TARSKI:def 1;
      then e1 SJoins {v},{w},G1 by A6, GLIB_000:17;
      hence contradiction by A2, GLIB_000:def 30;
    end;
    hence card G1.edgesBetween({v},{w})
      = card G2.edgesBetween({f.v},{f.w}) by A2;
  end;
  suppose G1.edgesBetween({v},{w}) <> {};
    then consider e1 being object such that
      A7: G1.edgesBetween({v},{w}) = {e1} by ZFMISC_1:131;
    set v1 = (the_Source_of G1).e1, v2 = (the_Target_of G1).e1;
    e1 in G1.edgesBetween({v},{w}) by A7, TARSKI:def 1;
    then e1 SJoins {v},{w},G1 by GLIB_000:def 30;
    then A8: e1 in the_Edges_of G1 &
      (v1 in {v} & v2 in {w} or v1 in {w} & v2 in {v}) by GLIB_000:def 15;
    then v1 = v & v2 = w or v1 = w & v2 = v by TARSKI:def 1;
    then A9: e1 Joins v,w,G1 by A8, GLIB_000:def 13;
    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 Joins f.v,f.w,G2 by A9, Th1;
    f.v in {f.v} & f.w in {f.w} by TARSKI:def 1;
    then e2 SJoins {f.v},{f.w},G2 by A10, GLIB_000:17;
    then e2 in G2.edgesBetween({f.v},{f.w}) by GLIB_000:def 30;
    then consider e being object such that
      A11: G2.edgesBetween({f.v},{f.w}) = {e} by ZFMISC_1:131;
    card G1.edgesBetween({v},{w}) = 1 &
      card G2.edgesBetween({f.v},{f.w}) = 1 by A7, A11, CARD_1:30;
    hence card G1.edgesBetween({v},{w})
      = card G2.edgesBetween({f.v},{f.w});
  end;
end;
