reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve L,M for Element of NAT;
reserve f for Function of A,B;
reserve f for Function;
reserve x1,x2,x3,x4,x5 for object;
reserve p for FinSequence;
reserve ND for non empty set;
reserve y1,y2,y3,y4,y5 for Element of ND;

theorem
  for Y being set, E1, E2 being Equivalence_Relation of Y st
  Class E1 = Class E2 holds E1 = E2
proof
  let Y be set, E1, E2 be Equivalence_Relation of Y such that
A1: Class E1 = Class E2;
  now
    let x be object;
    hereby
      assume
A2:   x in E1;
      then consider a, b being object such that
A3:   x = [a, b] and
A4:   a in Y and
A5:   b in Y by RELSET_1:2;
      reconsider a,b as Element of Y by A4,A5;
      Class (E1, b) in Class E2 by A1,A4,EQREL_1:def 3;
      then consider c being object such that
      c in Y and
A6:   Class (E1, b) = Class (E2, c) by EQREL_1:def 3;
      b in Class (E1, b) by A4,EQREL_1:20;
      then [b, c] in E2 by A6,EQREL_1:19;
      then
A7:   [c, b] in E2 by EQREL_1:6;
      a in Class (E1, b) by A2,A3,EQREL_1:19;
      then [a, c] in E2 by A6,EQREL_1:19;
      hence x in E2 by A3,A7,EQREL_1:7;
    end;
    assume
A8: x in E2;
    then consider a, b being object such that
A9: x = [a, b] and
A10: a in Y and
A11: b in Y by RELSET_1:2;
    reconsider a, b as Element of Y by A10,A11;
    Class (E2, b) in Class E1 by A1,A10,EQREL_1:def 3;
    then consider c being object such that
    c in Y and
A12: Class (E2, b) = Class (E1, c) by EQREL_1:def 3;
    b in Class (E2, b) by A10,EQREL_1:20;
    then [b, c] in E1 by A12,EQREL_1:19;
    then
A13: [c, b] in E1 by EQREL_1:6;
    a in Class (E2, b) by A8,A9,EQREL_1:19;
    then [a, c] in E1 by A12,EQREL_1:19;
    hence x in E1 by A9,A13,EQREL_1:7;
  end;
  hence thesis;
end;
