reserve A for non empty set,
  a for Element of A;
reserve A for set;
reserve B,C for Element of Fin DISJOINT_PAIRS A,
  x for Element of [:Fin A, Fin A:],
  a,b,c,d,s,t,s9,t9,t1,t2,s1,s2 for Element of DISJOINT_PAIRS A,
  u,v,w for Element of NormForm A;
reserve K,L for Element of Normal_forms_on A;

theorem Th4:
  for X being set st X c= K holds X in Normal_forms_on A
proof
  let X be set;
  assume
A1: X c= K;
  K c= DISJOINT_PAIRS A by FINSUB_1:def 5;
  then X c= DISJOINT_PAIRS A by A1;
  then reconsider B = X as Element of Fin DISJOINT_PAIRS A by A1,FINSUB_1:def 5
;
  for a,b st a in B & b in B & a c= b holds a = b by A1,NORMFORM:32;
  hence thesis;
end;
