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;

theorem Th33:
  Class EqR is a_partition of X
proof
  now
    let x be object;
    now
      consider Z such that
A1:   Z = Class(EqR,x);
      assume
A2:   x in X;
      then Z in Class EqR by A1,Def3;
      hence ex Z st x in Z & Z in Class EqR by A2,A1,Th20;
    end;
    hence x in X iff ex Z st x in Z & Z in Class EqR;
  end;
  hence union(Class EqR) = X by TARSKI:def 4;
  let A;
  assume A in Class EqR;
  then
A3: ex x st x in X & A = Class(EqR,x) by Def3;
  hence A <> {} by Th20;
  let B;
  assume B in Class EqR;
  then ex y st y in X & B = Class(EqR,y) by Def3;
  hence thesis by A3,Th24;
end;
