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 Th12:
  for R holds ex EqR st R c= EqR & for EqR2 st R c= EqR2 holds EqR c= EqR2
proof
  let R;
  defpred P[set] means $1 is Equivalence_Relation of X & R c= $1;
  consider F being Subset-Family of [:X,X:] such that
A1: for AX holds AX in F iff P[AX] from SUBSET_1:sch 3;
  R c= nabla X;
  then
A2: F <> {} by A1;
  for Y st Y in F holds Y is Equivalence_Relation of X by A1;
  then reconsider EqR = meet F as Equivalence_Relation of X by A2,Th11;
A3: for EqR2 st R c= EqR2 holds EqR c= EqR2 by A1,SETFAM_1:3;
  take EqR;
  for Y st Y in F holds R c= Y by A1;
  hence thesis by A2,A3,SETFAM_1:5;
end;
