reserve A for QC-alphabet;
reserve k,n,m for Nat;
reserve P for QC-pred_symbol of A;
reserve F for Element of QC-WFF(A);
reserve Q for QC-pred_symbol of A;
reserve F, G for (Element of QC-WFF(A)), s for FinSequence;
reserve p for Element of QC-WFF(A);
reserve F for Element of QC-WFF(A);

theorem Th18:
  (@VERUM(A).1)`1 = 0 & (F is atomic implies
   ex k being Nat st @F.1 is QC-pred_symbol of k, A)
  & (F is negative implies (@F.1)`1 = 1) &
(F is conjunctive implies (@F.1)`1 = 2) & (F is universal implies (@F.1)`1 = 3)
proof
  thus (@VERUM(A).1)`1 = [0,0]`1
    .= 0;
  thus F is atomic implies ex k being Nat st @F.1 is QC-pred_symbol
  of k, A
  proof
    assume F is atomic;
    then consider
    k being Nat, P being (QC-pred_symbol of k, A), ll being
    QC-variable_list of k, A such that
A1: F = P!ll;
    @F = <*P*>^ll by A1,Th8;
    then @F.1 = P by FINSEQ_1:41;
    hence thesis;
  end;
  thus F is negative implies (@F.1)`1 = 1
  proof
    assume F is negative;
    then ex p being Element of QC-WFF(A) st F = 'not' p;
    then @F.1 = [1, 0] by FINSEQ_1:41;
    hence thesis;
  end;
  thus F is conjunctive implies (@F.1)`1 = 2
  proof
    assume F is conjunctive;
    then consider p, q being Element of QC-WFF(A) such that
A2: F = p '&' q;
    @F = <*[2, 0]*>^(@p^@q) by A2,FINSEQ_1:32;
    then @F.1 = [2, 0] by FINSEQ_1:41;
    hence thesis;
  end;
  thus F is universal implies (@F.1)`1 = 3
  proof
    assume F is universal;
    then consider
    x being bound_QC-variable of A, p being Element of QC-WFF(A) such that
A3: F = All(x, p);
    @F = <*[3, 0]*>^(<*x*>^@p) by A3,FINSEQ_1:32;
    then @F.1 = [3, 0] by FINSEQ_1:41;
    hence thesis;
  end;
end;
