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;
reserve f,f9 for (Element of Funcs(DISJOINT_PAIRS A, [:Fin A,Fin A:])),
  g,h for Element of Funcs(DISJOINT_PAIRS A, [A]);

theorem Th18:
  c in B =>> C implies ex f st f.:B c= C & c = FinPairUnion(B,
  pair_diff A.:(f,incl DISJOINT_PAIRS A))
proof
  assume c in B =>> C;
  then
  c in { FinPairUnion(B,pair_diff A.:(f,incl DISJOINT_PAIRS A)) : f.:B c=
  C } by XBOOLE_0:def 4;
  then
  ex f st c = FinPairUnion(B,pair_diff A.:(f,incl DISJOINT_PAIRS A)) & f.:
  B c= C;
  hence thesis;
end;
