reserve p,q,r,s,A,B for Element of PL-WFF,
  F,G,H for Subset of PL-WFF,
  k,n for Element of NAT,
  f,f1,f2 for FinSequence of PL-WFF;
reserve M for PLModel;

theorem thaa:
  F |- A => A
  proof
    A=>((A=>A)=>A) in PL_axioms by withplax;then
A1: F |- A=>((A=>A)=>A) by th42;
    A=>(A=>A) in PL_axioms by withplax;then
A2: F |- A=>(A=>A) by th42;
    (A=>((A=>A)=>A)) => ((A=>(A=>A))=>(A=>A)) in PL_axioms by withplax;then
    F |- (A=>((A=>A)=>A)) => ((A=>(A=>A))=>(A=>A)) by th42;then
    F |- (A=>(A=>A))=>(A=>A) by th43,A1;
    hence F|- A=>A by th43,A2;
end;
