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 th42:
  p in PL_axioms or p in F implies F |- p
 proof
  defpred P1[set,set] means $2=p;
  A1: for k being Nat st k in Seg 1 holds ex x being Element of
  PL-WFF st P1[k,x];
  consider f such that
   A2: dom f=Seg 1 & for k being Nat st k in Seg 1 holds P1[k,f.k]
   from FINSEQ_1:sch 5(A1);
  A3: len f=1 by A2,FINSEQ_1:def 3;
  1 in Seg 1;
  then A4: f.1=p by A2;
  assume A5: p in PL_axioms or p in F;
  for j be Nat st 1<=j & j<=len f holds prc f,F,j
  proof
   let j be Nat;
   assume A6: 1<=j & j<=len f;
   per cases by A5;
   suppose p in PL_axioms;
    hence thesis by A3,A4,A6,XXREAL_0:1;
   end;
   suppose p in F;
    hence thesis by A3,A4,A6,XXREAL_0:1;
   end;
  end;
  hence F|-p by A3,A4;
 end;
