
theorem Th73:
  for G being _Graph, E2 being RepEdgeSelection of G
  ex E1 being RepDEdgeSelection of G st E2 c= E1
proof
  let G be _Graph, E2 be RepEdgeSelection of G;
  set A = {{e where e is Element of the_Edges_of G : e DJoins v1,v2,G}
    where v1,v2 is Vertex of G : (ex e0 being object st e0 DJoins v1,v2,G) &
      (for e0 being object st e0 DJoins v1,v2,G holds not e0 in E2)};
  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 DJoins v1,v2,G} and
      A4: ex e0 being object st e0 DJoins v1,v2,G and
      for e0 being object st e0 DJoins v1,v2,G holds not e0 in E2;
    reconsider B = x as set by A3;
    consider e0 being object such that
      A5: e0 DJoins v1,v2,G by A4;
    reconsider e0 as Element of the_Edges_of G by A5, GLIB_000:def 14;
    e0 in B by A3, A5;
    then reconsider B as non empty set;
    take the Element of B, B;
    thus thesis;
  end;
  consider f being Function such that
    A6: 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 the_Edges_of G
  proof
    let e be object;
    assume e in rng f;
    then consider C being object such that
      A7: C in dom f & f.C = e by FUNCT_1:def 3;
    consider C0 being non empty set such that
      A8: C = C0 & f.C = the Element of C0 by A6, A7;
    consider v1,v2 being Vertex of G such that
      A9: C = {e where e is Element of the_Edges_of G : e DJoins v1,v2,G} and
      ex e0 being object st e0 DJoins v1,v2,G and
      for e0 being object st e0 DJoins v1,v2,G holds not e0 in E2 by A6, A7;
    e in C0 by A7, A8;
    then consider e2 being Element of the_Edges_of G such that
      A10: e = e2 & e2 DJoins v1,v2,G by A8, A9;
    thus e in the_Edges_of G by A10, GLIB_000:def 14;
  end;
  then rng f c= the_Edges_of G by TARSKI:def 3;
  then reconsider E1 = E2 \/ rng f as Subset of the_Edges_of G by XBOOLE_1:8;
  for v,w,e0 being object st e0 DJoins v,w,G
    ex e being object st e DJoins v,w,G & e in E1 &
     for e9 being object st e9 DJoins v,w,G & e9 in E1 holds e9 = e
  proof
    let v,w,e0 be object;
    assume A11: e0 DJoins v,w,G;
    then e0 Joins v,w,G by GLIB_000:16;
    then consider e2 being object such that
      A12: e2 Joins v,w,G & e2 in E2 and
      A13: for e8 being object st e8 Joins v,w,G & e8 in E2 holds e8 = e2
      by Def5;
    per cases by A12, GLIB_000:16;
    suppose A14: e2 DJoins v,w,G;
      take e2;
      thus e2 DJoins v,w,G & e2 in E1 by A12, A14, TARSKI:def 3, XBOOLE_1:7;
      let e9 be object;
      assume A15: e9 DJoins v,w,G & e9 in E1;
      not e9 in rng f
      proof
        assume e9 in rng f;
        then consider C being object such that
          A16: C in dom f & f.C = e9 by FUNCT_1:def 3;
        consider C0 being non empty set such that
          A17: C = C0 & f.C = the Element of C0 by A6, A16;
        consider v1,v2 being Vertex of G such that
          A18: C = {e where e is Element of the_Edges_of G : e DJoins v1,v2,G}
          and ex e0 being object st e0 DJoins v1,v2,G and
          A19: for e0 being object st e0 DJoins v1,v2,G holds not e0 in E2
          by A6, A16;
        e9 in C0 by A16, A17;
        then consider e being Element of the_Edges_of G such that
          A20: e9 = e & e DJoins v1,v2,G by A17, A18;
        v = v1 & w = v2 by A15, A20, GLIB_000:125;
        hence contradiction by A12, A14, A19;
      end;
      then A21: e9 in E2 by A15, XBOOLE_0:def 3;
      e9 Joins v,w,G by A15, GLIB_000:16;
      hence e9 = e2 by A13, A21;
    end;
    suppose A22: e2 DJoins w,v,G & not e2 DJoins v,w,G;
      set B = {e where e is Element of the_Edges_of G : e DJoins v,w,G};
      A23: for e9 being object st e9 DJoins v,w,G holds not e9 in E2
      proof
        given e9 being object such that
          A24: e9 DJoins v,w,G & e9 in E2;
        e9 Joins v,w,G by A24, GLIB_000:16;
        hence contradiction by A13, A22, A24;
      end;
      v in the_Vertices_of G & w in the_Vertices_of G by A12, GLIB_000:13;
      then A25: B in A by A11, A23;
      then consider B0 being non empty set such that
        A26: B = B0 & f.B = the Element of B0 by A6;
      f.B in B0 by A26;
      then consider e being Element of the_Edges_of G such that
        A27: f.B = e & e DJoins v,w,G by A26;
      take e;
      thus e DJoins v,w,G by A27;
      e in rng f by A6, A25, A27, FUNCT_1:3;
      hence e in E1 by XBOOLE_1:7, TARSKI:def 3;
      let e9 be object;
      assume A28: e9 DJoins v,w,G & e9 in E1;
      not e9 in E2
      proof
        assume A29: e9 in E2;
        e9 Joins v,w,G by A28, GLIB_000:16;
        hence contradiction by A13, A22, A28, A29;
      end;
      then e9 in rng f by A28, XBOOLE_0:def 3;
      then consider C being object such that
        A30: C in dom f & f.C = e9 by FUNCT_1:def 3;
      consider C0 being non empty set such that
        A31: C = C0 & f.C = the Element of C0 by A6, A30;
      consider v1,v2 being Vertex of G such that
        A32: C = {e1 where e1 is Element of the_Edges_of G : e1 DJoins v1,v2,G}
        and ex e7 being object st e7 DJoins v1,v2,G and
        for e7 being object st e7 DJoins v1,v2,G holds not e7 in E2
        by A6, A30;
      e9 in C0 by A30, A31;
      then consider e1 being Element of the_Edges_of G such that
        A33: e9 = e1 & e1 DJoins v1,v2,G by A31, A32;
      v = v1 & w = v2 by A28, A33, GLIB_000:125;
      hence e9 = e by A27, A30, A32;
    end;
  end;
  then reconsider E1 as RepDEdgeSelection of G by Def6;
  take E1;
  thus thesis by XBOOLE_1:7;
end;
