reserve G for _Graph;

theorem
  card Class EdgeAdjEqRel(G) c= card VertexAdjSymRel(G)
proof
  set R = VertexAdjSymRel(G);
  defpred P[object,object] means ex e being object st
    e Joins $1`1,$1`2,G & $2 = Class(EdgeAdjEqRel(G),e);
  A1: for x,y1,y2 being object st x in R & P[x,y1] & P[x,y2] holds y1 = y2
  proof
    let x,y1,y2 be object;
    assume A2: x in R & P[x,y1] & P[x,y2];
    then consider e1 being object such that
      A3: e1 Joins x`1,x`2,G & y1 = Class(EdgeAdjEqRel(G),e1);
    consider e2 being object such that
      A4: e2 Joins x`1,x`2,G & y2 = Class(EdgeAdjEqRel(G),e2) by A2;
    [e1,e2] in EdgeAdjEqRel(G) by A3, A4, GLIB_009:def 3;
    then A5: e2 in Class(EdgeAdjEqRel(G),e1) by EQREL_1:18;
    e1 in the_Edges_of G by A3, GLIB_000:def 13;
    hence thesis by A3, A4, A5, EQREL_1:23;
  end;
  A6: for x being object st x in R ex y being object st P[x,y]
  proof
    let x be object;
    assume A7: x in R;
    then consider v,w being object such that
      A8: x = [v,w] by RELAT_1:def 1;
    consider e being object such that
      A9: e Joins v,w,G by A7, A8, Th32;
    take Class(EdgeAdjEqRel(G),e),e;
    thus thesis by A8, A9;
  end;
  consider f being Function such that
    A10: dom f = R & for x being object st x in R holds P[x,f.x]
    from FUNCT_1:sch 2(A1,A6);
  now
    let y be object;
    hereby
      assume y in rng f;
      then consider x being object such that
        A11: x in dom f & f.x = y by FUNCT_1:def 3;
      consider e being object such that
        A12: e Joins x`1,x`2,G & y = Class(EdgeAdjEqRel(G),e) by A10, A11;
      e in the_Edges_of G by A12, GLIB_000:def 13;
      hence y in Class EdgeAdjEqRel(G) by A12, EQREL_1:def 3;
    end;
    assume y in Class EdgeAdjEqRel(G);
    then consider e being object such that
      A13: e in the_Edges_of G & y = Class(EdgeAdjEqRel(G),e) by EQREL_1:def 3;
    set x = [(the_Source_of G).e,(the_Target_of G).e];
    A14: e Joins x`1,x`2,G by A13, GLIB_000:def 13;
    then A15: x in R by Th32;
    then P[x,f.x] by A10;
    then f.x = y by A1, A13, A14, A15;
    hence y in rng f by A10, A15, FUNCT_1:3;
  end;
  then rng f = Class EdgeAdjEqRel(G) by TARSKI:2;
  hence thesis by A10, CARD_1:12;
end;
