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;

theorem plhp:
  PL-WFF c= HP-WFF
  proof
    let x be object;
    assume
A0: x in PL-WFF;
    defpred P[Element of PL-WFF] means $1 in HP-WFF;
    VERUM = FaLSUM;
    :: syntactically only,since represented by identical sequences
    then A1: P[FaLSUM];
A2: for n holds P[Prop n]
    proof
      let n;
      Prop n = prop n;
      hence thesis;
    end;
A3: for r,s st P[r] & P[s] holds P[r => s]
    proof
      let r,s;
      assume P[r] & P[s];then
      reconsider r1 = r, s1 = s as Element of HP-WFF;
      r1 => s1 in HP-WFF;
      hence P[r => s];
    end;
A4: for A holds P[A] from PLInd(A1,A2,A3);
    reconsider x1 = x as Element of PL-WFF by A0;
    x1 in HP-WFF by A4;
    hence x in HP-WFF;
  end;
