theorem Th88:
  PA1 is_finer_than PA2 implies for p2 being Element of PA2
  ex p1 being Element of PA1 st p1 c= p2
proof
  assume
A1: PA1 is_finer_than PA2;
  let p2 be Element of PA2;
  consider E1 being Equivalence_Relation of A such that
A2: PA1 = Class E1 by EQREL_1:34;
  reconsider p29 = p2 as Subset of A;
  consider E2 being Equivalence_Relation of A such that
A3: PA2 = Class E2 by EQREL_1:34;
  consider a being object such that
A4: a in A and
A5: p29 = Class (E2, a) by A3,EQREL_1:def 3;
A6: a in Class (E1, a) by A4,EQREL_1:20;
  reconsider E1a = Class (E1, a) as Element of PA1 by A2,A4,EQREL_1:def 3;
  consider p22 being set such that
A7: p22 in PA2 and
A8: E1a c= p22 by A1,SETFAM_1:def 2;
  reconsider p229 = p22 as Subset of A by A7;
  take E1a;
  ex b being object st b in A & p229 = Class (E2, b) by A3,A7,EQREL_1:def 3;
  hence thesis by A5,A8,A6,EQREL_1:23;
end;
