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 Th32:
  for PA being a_partition of Y holds %O(Y) '>' PA & PA '>' %I(Y)
proof
  let PA be a_partition of Y;
A1: for a being set st a in PA ex b being set st b in %O(Y) & a c= b
  proof
    let a be set;
    assume
A2: a in PA; then
A3: a c= Y;
A4: a<>{} by A2,EQREL_1:def 4;
    set x = the Element of a;
A5: x in Y by A2,A4,TARSKI:def 3;
    union %O(Y) = Y by EQREL_1:def 4;
    then consider b being set such that
    x in b and
A6: b in %O(Y) by A5,TARSKI:def 4;
    a c= b by A3,A6,TARSKI:def 1;
    hence thesis by A6;
  end;
  for a being set st a in %I(Y) ex b being set st b in PA & a c= b
  proof
    let a be set;
    assume
A7: a in %I(Y);
then  a in the set of all {x} where x is Element of Y by EQREL_1:37;
    then consider x be Element of Y such that
A8: a={x};
A9: a<>{} by A7,EQREL_1:def 4;
    set u = the Element of a;
A10: u=x by A8,TARSKI:def 1;
A11: u in Y by A7,A9,TARSKI:def 3;
    union PA = Y by EQREL_1:def 4;
    then consider b being set such that
A12: u in b and
A13: b in PA by A11,TARSKI:def 4;
    a c= b by A8,A10,A12,TARSKI:def 1;
    hence thesis by A13;
  end;
  hence thesis by A1,SETFAM_1:def 2;
end;
