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 Th25:
  @u ^ { a } = {} implies Atom(A).a [= pseudo_compl(A).u
proof
  assume
A1: @u ^ { a } = {};
  now
    let c;
    assume c in Atom(A).a;
    then c = a by Th6;
    then consider b such that
A2: b in -@u and
A3: b c= c by A1,Th19;
    consider d such that
A4: d c= b and
A5: d in mi(-@u) by A2,NORMFORM:41;
    take e = d;
    thus e in pseudo_compl(A).u by A5,Def8;
    thus e c= c by A3,A4,NORMFORM:2;
  end;
  hence thesis by Lm3;
end;
