
theorem Th72:
  for G being _Graph, E1 being RepDEdgeSelection of G
  ex E2 being RepEdgeSelection of G st E2 c= E1
proof
  let G be _Graph, E1 be RepDEdgeSelection of G;
  set A = {{e where e is Element of the_Edges_of G : e Joins v1,v2,G & e in E1}
    where v1,v2 is Vertex of G : ex e0 being object st e0 Joins v1,v2,G};
  defpred P[object,object] means ex S being non empty set
    st $1 = S & $2 = the Element of S;
  A1: for x,y1,y2 being object st x in A & P[x,y1] & P[x,y2] holds y1 = y2;
  A2: for x being object st x in A ex y being object st P[x,y]
  proof
    let x be object;
    assume x in A;
    then consider v1,v2 being Vertex of G such that
      A3: x = {e where e is Element of the_Edges_of G :
        e Joins v1,v2,G & e in E1} and
      A4: ex e0 being object st e0 Joins v1,v2,G;
    reconsider B = x as set by A3;
    consider e0 being object such that
      A5: e0 Joins v1,v2,G by A4;
    per cases by A5, GLIB_000:16;
    suppose e0 DJoins v1,v2,G;
      then consider e being object such that
        A6: e DJoins v1,v2,G & e in E1 and
        for e9 being object st e9 DJoins v1,v2,G & e9 in E1 holds e9 = e
        by Def6;
      e in the_Edges_of G & e Joins v1,v2,G by A6, GLIB_000:16;
      then e in B by A3, A6;
      then reconsider B as non empty set;
      take the Element of B, B;
      thus thesis;
    end;
    suppose e0 DJoins v2,v1,G;
      then consider e being object such that
        A7: e DJoins v2,v1,G & e in E1 and
        for e9 being object st e9 DJoins v2,v1,G & e9 in E1 holds e9 = e
        by Def6;
      e in the_Edges_of G & e Joins v1,v2,G by A7, GLIB_000:16;
      then e in B by A3, A7;
      then reconsider B as non empty set;
      take the Element of B, B;
      thus thesis;
    end;
  end;
  consider f being Function such that
    A8: dom f = A & for x being object st x in A holds P[x,f.x]
    from FUNCT_1:sch 2(A1,A2);
  for e being object holds e in rng f implies e in E1
  proof
    let e be object;
    assume e in rng f;
    then consider C being object such that
      A9: C in dom f & f.C = e by FUNCT_1:def 3;
    consider C0 being non empty set such that
      A10: C = C0 & f.C = the Element of C0 by A8, A9;
    consider v1,v2 being Vertex of G such that
      A11: C = {e2 where e2 is Element of the_Edges_of G :
        e2 Joins v1,v2,G & e2 in E1} and
      ex e8 being object st e8 Joins v1,v2,G by A8, A9;
    e in C0 by A9, A10;
    then consider e2 being Element of the_Edges_of G such that
      A12: e = e2 & e2 Joins v1,v2,G & e2 in E1 by A10, A11;
    thus e in E1 by A12;
  end;
  then A13: rng f c= E1 by TARSKI:def 3;
  then reconsider E2 = rng f as Subset of the_Edges_of G by XBOOLE_1:1;
  for v,w,e0 being object st e0 Joins v,w,G
    ex e being object st e Joins v,w,G & e in E2 &
     for e9 being object st e9 Joins v,w,G & e9 in E2 holds e9 = e
  proof
    let v,w,e0 be object;
    assume A14: e0 Joins v,w,G;
    set B = {e where e is Element of the_Edges_of G :
      e Joins v,w,G & e in E1};
    v in the_Vertices_of G & w in the_Vertices_of G by A14, GLIB_000:13;
    then A15: B in A by A14;
    then consider B0 being non empty set such that
      A16: B = B0 & f.B = the Element of B0 by A8;
    f.B in B by A16;
    then consider e being Element of the_Edges_of G such that
      A17: f.B = e & e Joins v,w,G & e in E1;
    take e;
    thus e Joins v,w,G by A17;
    thus e in E2 by A8, A15, A17, FUNCT_1:3;
    let e9 be object;
    assume A18: e9 Joins v,w,G & e9 in E2;
    then consider C being object such that
      A19: C in dom f & f.C = e9 by FUNCT_1:def 3;
    consider v1,v2 being Vertex of G such that
      A20: C = {e2 where e2 is Element of the_Edges_of G :
        e2 Joins v1,v2,G & e2 in E1} and
      ex e8 being object st e8 Joins v1,v2,G by A8, A19;
    consider C0 being non empty set such that
      A21: C = C0 & f.C = the Element of C0 by A8, A19;
    e9 in C0 by A19, A21;
    then consider e2 being Element of the_Edges_of G such that
      A22: e9 = e2 & e2 Joins v1,v2,G & e2 in E1 by A20, A21;
    per cases by A18, A22, GLIB_000:15;
    suppose v = v1 & w = v2;
      hence e9 = e by A17, A19, A20;
    end;
    suppose A23: v = v2 & w = v1;
      for x being object holds x in C0 iff x in B
      proof
        let x be object;
        hereby
          assume x in C0;
          then consider e2 being Element of the_Edges_of G such that
            A24: x = e2 & e2 Joins v1,v2,G & e2 in E1 by A20, A21;
          e2 Joins v,w,G by A23, A24, GLIB_000:14;
          hence x in B by A24;
        end;
        assume x in B;
        then consider e1 being Element of the_Edges_of G such that
          A25: x = e1 & e1 Joins v,w,G & e1 in E1;
        e1 Joins v1,v2,G by A23, A25, GLIB_000:14;
        hence x in C0 by A20, A21, A25;
      end;
      hence e9 = e by A17, A19, A21, TARSKI:2;
    end;
  end;
  then reconsider E2 as RepEdgeSelection of G by Def5;
  take E2;
  thus thesis by A13;
end;
