
theorem Th80:
  for G1 being _Graph, G2 being Subgraph of G1,E2 being RepEdgeSelection of G2
  ex E1 being RepEdgeSelection of G1 st E2 = E1 /\ the_Edges_of G2
proof
  let G1 be _Graph, G2 be Subgraph of G1, E2 be RepEdgeSelection of G2;
  set A = {{e where e is Element of the_Edges_of G1 : e Joins v1,v2,G1}
    where v1,v2 is Vertex of G1 : (ex e0 being object st e0 Joins v1,v2,G1) &
      (for e0 being object st e0 Joins v1,v2,G1 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 G1 such that
      A3: x = {e where e is Element of the_Edges_of G1 : e Joins v1,v2,G1} and
      A4: ex e0 being object st e0 Joins v1,v2,G1 and
      for e0 being object st e0 Joins v1,v2,G1 holds not e0 in E2;
    reconsider B = x as set by A3;
    consider e0 being object such that
      A5: e0 Joins v1,v2,G1 by A4;
    reconsider e0 as Element of the_Edges_of G1 by A5, GLIB_000:def 13;
    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 G1
  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 G1 such that
      A9: C = {e where e is Element of the_Edges_of G1 : e Joins v1,v2,G1} and
      ex e0 being object st e0 Joins v1,v2,G1 and
      for e0 being object st e0 Joins v1,v2,G1 holds not e0 in E2 by A6, A7;
    e in C0 by A7, A8;
    then consider e2 being Element of the_Edges_of G1 such that
      A10: e = e2 & e2 Joins v1,v2,G1 by A8, A9;
    thus e in the_Edges_of G1 by A10, GLIB_000:def 13;
  end;
  then A11: rng f c= the_Edges_of G1 by TARSKI:def 3;
  the_Edges_of G2 c= the_Edges_of G1;
  then E2 c= the_Edges_of G1 by XBOOLE_1:1;
  then reconsider E1 = E2 \/ rng f as Subset of the_Edges_of G1
    by A11, XBOOLE_1:8;
  for v,w,e0 being object st e0 Joins v,w,G1
    ex e being object st e Joins v,w,G1 & e in E1 &
     for e9 being object st e9 Joins v,w,G1 & e9 in E1 holds e9 = e
  proof
    let v,w,e0 be object;
    A12: v is set & w is set by TARSKI:1;
    assume A13: e0 Joins v,w,G1;
    per cases;
    suppose ex e1 being object st e1 Joins v,w,G1 & e1 in E2;
      then consider e1 being object such that
        A14: e1 Joins v,w,G1 & e1 in E2;
      e1 Joins v,w,G2 by A12, A14, GLIB_000:73;
      then consider e being object such that
        A15: e Joins v,w,G2 & e in E2 and
        A16: for e8 being object st e8 Joins v,w,G2 & e8 in E2 holds e8 = e
        by Def5;
      take e;
      thus A17: e Joins v,w,G1 by A12, A15, GLIB_000:72;
      thus e in E1 by A15, XBOOLE_0:def 3;
      let e9 be object;
      assume A18: e9 Joins v,w,G1 & e9 in E1;
      not e9 in rng f
      proof
        assume e9 in rng f;
        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 G1 such that
          A20: C = {k where k is Element of the_Edges_of G1 : k Joins v1,v2,G1}
          and ex k0 being object st k0 Joins v1,v2,G1 and
          A21: for k0 being object st k0 Joins v1,v2,G1 holds not k0 in E2
          by A6, A19;
        consider C0 being non empty set such that
          A22: C = C0 & f.C = the Element of C0 by A6, A19;
        e9 in C0 by A19, A22;
        then consider k being Element of the_Edges_of G1 such that
          A23: e9 = k & k Joins v1,v2,G1 by A20, A22;
        v1 = v & v2 = w or v1 = w & v2 = v by A18, A23, GLIB_000:15;
        hence contradiction by A15, A17, A21, GLIB_000:14;
      end;
      then A24: e9 in E2 by A18, XBOOLE_0:def 3;
      then e9 Joins v,w,G2 by A12, A18, GLIB_000:73;
      hence e9 = e by A16, A24;
    end;
    suppose A25: for e1 being object st e1 Joins v,w,G1 holds not e1 in E2;
      A26: v is Vertex of G1 & w is Vertex of G1 by A13, GLIB_000:13;
      set B = {e where e is Element of the_Edges_of G1 : e Joins v,w,G1};
      A27: B in A by A13, A25, A26;
      then consider B0 being non empty set such that
        A28: B = B0 & f.B = the Element of B0 by A6;
      f.B in B by A28;
      then consider e being Element of the_Edges_of G1 such that
        A29: f.B = e & e Joins v,w,G1;
      take e;
      thus e Joins v,w,G1 by A29;
      e in rng f by A6, A27, A29, FUNCT_1:3;
      hence e in E1 by XBOOLE_0:def 3;
      let e9 be object;
      assume A30: e9 Joins v,w,G1 & e9 in E1;
      then not e9 in E2 by A25;
      then e9 in rng f by A30, XBOOLE_0:def 3;
      then consider C being object such that
        A31: C in dom f & f.C = e9 by FUNCT_1:def 3;
      consider v1,v2 being Vertex of G1 such that
        A32: C = {k where k is Element of the_Edges_of G1 : k Joins v1,v2,G1}
        and ex k0 being object st k0 Joins v1,v2,G1 and
        for k0 being object st k0 Joins v1,v2,G1 holds not k0 in E2
        by A6, A31;
      consider C0 being non empty set such that
        A33: C = C0 & f.C = the Element of C0 by A6, A31;
      f.C in C0 by A33;
      then consider k being Element of the_Edges_of G1 such that
        A34: f.C = k & k Joins v1,v2,G1 by A32, A33;
      for x being object holds x in B iff x in C0
      proof
        let x be object;
        A35: v1 = v & v2 = w or v1 = w & v2 = v by A30, A31, A34, GLIB_000:15;
        hereby
          assume x in B;
          then consider k being Element of the_Edges_of G1 such that
            A36: x = k & k Joins v,w,G1;
          k Joins v1,v2,G1 by A35, A36, GLIB_000:14;
          hence x in C0 by A32, A33, A36;
        end;
        assume x in C0;
        then consider k being Element of the_Edges_of G1 such that
          A37: x = k & k Joins v1,v2,G1 by A32, A33;
        k Joins v,w,G1 by A35, A37, GLIB_000:14;
        hence x in B by A37;
      end;
      hence e9 = e by A29, A31, A33, TARSKI:2;
    end;
  end;
  then reconsider E1 as RepEdgeSelection of G1 by Def5;
  take E1;
  for x being object holds x in E2 iff x in E1 & x in the_Edges_of G2
  proof
    let x be object;
    thus x in E2 implies x in E1 & x in the_Edges_of G2
      by TARSKI:def 3, XBOOLE_1:7;
    assume A38: x in E1 & x in the_Edges_of G2;
    not x in rng f
    proof
      assume x in rng f;
      then consider C being object such that
        A39: C in dom f & f.C = x by FUNCT_1:def 3;
      consider v1,v2 being Vertex of G1 such that
        A40: C = {e where e is Element of the_Edges_of G1 : e Joins v1,v2,G1}
        and ex e0 being object st e0 Joins v1,v2,G1 and
        A41: for e0 being object st e0 Joins v1,v2,G1 holds not e0 in E2
        by A6, A39;
      consider C0 being non empty set such that
        A42: C = C0 & f.C = the Element of C0 by A6, A39;
      f.C in C0 by A42;
      then consider e being Element of the_Edges_of G1 such that
        A43: f.C = e & e Joins v1,v2,G1 by A40, A42;
      x Joins v1,v2,G2 by A38, A39, A43, GLIB_000:73;
      then consider e1 being object such that
        A44: e1 Joins v1,v2,G2 & e1 in E2 and
        for e9 being object st e9 Joins v1,v2,G2 & e9 in E2 holds e9 = e1
        by Def5;
      thus contradiction by A41, A44, GLIB_000:72;
    end;
    hence thesis by A38, XBOOLE_0:def 3;
  end;
  hence thesis by XBOOLE_0:def 4;
end;
