reserve V, C, x, a, b for set;
reserve A, B for Element of SubstitutionSet (V, C);
reserve C for finite set;
reserve A, B for Element of SubstitutionSet (V, C);
reserve u, v for Element of SubstLatt (V, C);
reserve s, t, a, b for Element of PFuncs (V,C);
reserve K, L for Element of SubstitutionSet (V, C);

theorem Th22:
  for a being Element of PFuncs (V, C) st K = {a} & L = u & L ^ K
  = {} holds Atom(V, C).a [= pseudo_compl(V, C).u
proof
  let a be Element of PFuncs (V, C);
  assume that
A1: K = {a} and
A2: L = u and
A3: L ^ K = {};
  a in K by A1,TARSKI:def 1;
  then
A4: a is finite by Th1;
  now
    let c be set;
    assume
A5: c in Atom(V, C).a;
    then reconsider c9 = c as Element of PFuncs (V, C) by A4,Th21;
    c = a by A4,A5,Th21;
    then consider b be finite set such that
A6: b in -L and
A7: b c= c9 by A1,A3,Th14;
    consider d be set such that
A8: d c= b and
A9: d in mi(-L) by A6,SUBSTLAT:10;
    take e = d;
    thus e in pseudo_compl(V, C).u by A2,A9,Def4;
    thus e c= c by A7,A8;
  end;
  hence thesis by Lm8;
end;
