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;
