
theorem Th85:
  for G being _Graph, E1, E2 being RepEdgeSelection of G
  ex f being one-to-one Function st dom f = E1 & rng f = E2 &
    for e,v,w being object st e in E1 holds e Joins v,w,G iff f.e Joins v,w,G
proof
  let G be _Graph, E1, E2 be RepEdgeSelection of G;
  defpred P[object,object] means $2 in E2 & ex v,w being object
    st $1 Joins v,w,G & $2 Joins v,w,G;
  A1: for x,y1,y2 being object st x in E1 & P[x,y1] & P[x,y2] holds y1 = y2
  proof
    let x,y1,y2 be object;
    assume A2: x in E1 & P[x,y1] & P[x,y2];
    then consider v1,w1 being object such that
      A3: x Joins v1,w1,G & y1 Joins v1,w1,G;
    consider v2,w2 being object such that
      A4: x Joins v2,w2,G & y2 Joins v2,w2,G by A2;
    consider e2 being object such that
      e2 Joins v1,w1,G & e2 in E2 and
      A5: for e9 being object st e9 Joins v1,w1,G & e9 in E2 holds e9 = e2
      by A3, Def5;
    A6: y1 = e2 by A2, A3, A5;
    v1 = v2 & w1 = w2 or v1 = w2 & w1 = v2 by A3, A4, GLIB_000:15;
    hence y1 = y2 by A2, A4, A5, A6, GLIB_000:14;
  end;
  A7: for x being object st x in E1 ex y being object st P[x,y]
  proof
    let x be object;
    set v = (the_Source_of G).x, w = (the_Target_of G).x;
    assume x in E1;
    then A8: x Joins v,w,G by GLIB_000:def 13;
    then consider e2 being object such that
      A9: e2 Joins v,w,G & e2 in E2 and
      for e9 being object st e9 Joins v,w,G & e9 in E2 holds e9 = e2 by Def5;
    take e2;
    thus e2 in E2 by A9;
    take v,w;
    thus thesis by A8, A9;
  end;
  consider f being Function such that
    A10: dom f = E1 & for x being object st x in E1 holds P[x,f.x]
    from FUNCT_1:sch 2(A1,A7);
  now
    let x1,x2 be object;
    assume A11: x1 in dom f & x2 in dom f & f.x1 = f.x2;
    then consider v1,w1 being object such that
      A12: x1 Joins v1,w1,G & f.x1 Joins v1,w1,G by A10;
    consider v2,w2 being object such that
      A13: x2 Joins v2,w2,G & f.x2 Joins v2,w2,G by A10, A11;
    consider e1 being object such that
      e1 Joins v1,w1,G & e1 in E1 and
      A14: for e9 being object st e9 Joins v1,w1,G & e9 in E1 holds e9 = e1
      by A12, Def5;
    A15: x1 = e1 by A10, A11, A12, A14;
    v1 = v2 & w1 = w2 or v1 = w2 & w1 = v2 by A11, A12, A13, GLIB_000:15;
    hence x1 = x2 by A10, A11, A13, A14, A15, GLIB_000:14;
  end;
  then reconsider f as one-to-one Function by FUNCT_1:def 4;
  take f;
  thus dom f = E1 by A10;
  now
    let y be object;
    hereby
      assume y in rng f;
      then consider x being object such that
        A16: x in dom f & f.x = y by FUNCT_1:def 3;
      thus y in E2 by A10, A16;
    end;
    assume A17: y in E2;
    set v = (the_Source_of G).y, w = (the_Target_of G).y;
    A18: y Joins v,w,G by A17, GLIB_000:def 13;
    then consider e1 being object such that
      A19: e1 Joins v,w,G & e1 in E1 and
      for e9 being object st e9 Joins v,w,G & e9 in E1 holds e9 = e1
      by Def5;
    consider e2 being object such that
      e2 Joins v,w,G & e2 in E2 and
      A20: for e9 being object st e9 Joins v,w,G & e9 in E2 holds e9 = e2
      by A18, Def5;
    consider v0,w0 being object such that
      A21: e1 Joins v0,w0,G & f.e1 Joins v0,w0,G by A10, A19;
    v0 = v & w0 = w or v0 = w & w0 = v by A19, A21, GLIB_000:15;
    then f.e1 Joins v,w,G by A21, GLIB_000:14;
    then A22: f.e1 = e2 by A10, A19, A20;
    y = e2 by A17, A18, A20;
    hence y in rng f by A10, A19, A22, FUNCT_1:3;
  end;
  hence rng f = E2 by TARSKI:2;
  let e,v,w be object;
  assume e in E1;
  then consider v0,w0 being object such that
    A23: e Joins v0,w0,G & f.e Joins v0,w0,G by A10;
  hereby
    assume e Joins v,w,G;
    then v0 = v & w0 = w or v0 = w & w0 = v by A23, GLIB_000:15;
    hence f.e Joins v,w,G by A23, GLIB_000:14;
  end;
  assume f.e Joins v,w,G;
  then v0 = v & w0 = w or v0 = w & w0 = v by A23, GLIB_000:15;
  hence e Joins v,w,G by A23, GLIB_000:14;
end;
