reserve Y,Z for non empty set;
reserve PA,PB for a_partition of Y;
reserve A,B for Subset of Y;
reserve i,j,k for Nat;
reserve x,y,z,x1,x2,y1,z0,X,V,a,b,d,t,SFX,SFY for set;

theorem
  for X being set
  for P being a_partition of X holds P = Class ERl P
proof
  let X be set;
  let P be a_partition of X;
    set R = ERl P;
    now
      let A be Subset of X;
A13:  now
        set x = the Element of A;
        assume
A14:    A in P; then
A15:    A <> {} by EQREL_1:def 4;
        then
A16:    x in X by TARSKI:def 3;
        now
          let y be object;
A17:      now
            assume y in Class(R,x);
            then [y,x] in R by EQREL_1:19;
            then consider B being Subset of X such that
A18:        B in P & y in B and
A19:        x in B by Def6;
            A meets B by A15,A19,XBOOLE_0:3;
            hence y in A by A14,A18,EQREL_1:def 4;
          end;
          now
            assume y in A;
            then [y,x] in R by Def6,A14;
            hence y in Class(R,x) by EQREL_1:19;
          end;
          hence y in A iff y in Class(R,x) by A17;
        end;
        then A = Class(R,x) by TARSKI:2;
        hence A in Class R by A16,EQREL_1:def 3;
      end;
      now
        assume A in Class R;
        then consider x being object such that
A20:    x in X and
A21:    A = Class(R,x) by EQREL_1:def 3;
        x in Class(R,x) by A20,EQREL_1:20;
        then [x,x] in R by EQREL_1:19;
        then consider B being Subset of X such that
A22:    B in P and
A23:    x in B and
        x in B by Def6;
        now
          let y be object;
A24:      now
            assume y in A;
            then [y,x] in R by A21,EQREL_1:19;
            then consider C being Subset of X such that
A25:        C in P & y in C and
A26:        x in C by Def6;
            B meets C by A23,A26,XBOOLE_0:3;
            hence y in B by A22,A25,EQREL_1:def 4;
          end;
          now
            assume y in B;
            then [y,x] in R by Def6,A22,A23;
            hence y in A by A21,EQREL_1:19;
          end;
          hence y in A iff y in B by A24;
        end;
        hence A in P by A22,TARSKI:2;
      end;
      hence A in P iff A in Class R by A13;
    end;
    hence thesis by SETFAM_1:31;
end;
