reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;
reserve X for non empty set,
  x for Element of X;

theorem Th38:
  for S1,S2 being a_partition of X st (for x being Element of X
  holds EqClass(x,S1) = EqClass(x,S2)) holds S1=S2
proof
  let S1,S2 be a_partition of X;
  assume
A1: for x being Element of X holds EqClass(x,S1) = EqClass(x,S2);
  now
    let P be Subset of X;
    thus P in S1 implies P in S2
    proof
      set x = the Element of P;
      assume
A2:   P in S1;
      then
A3:   P is non empty by Def4;
      then x in P;
      then reconsider x as Element of X;
      P = EqClass(x,S1) by A2,A3,Def6;
      then P = EqClass(x,S2) by A1;
      hence thesis by Def6;
    end;
    thus P in S2 implies P in S1
    proof
      set x = the Element of P;
      assume
A4:   P in S2;
      then
A5:   P <> {} by Def4;
      then x in P;
      then reconsider x as Element of X;
      P = EqClass(x,S2) by A4,A5,Def6;
      then P = EqClass(x,S1) by A1;
      hence thesis by Def6;
    end;
  end;
  hence thesis by SETFAM_1:31;
end;
