reserve A for QC-alphabet;
reserve i,j,k,l,m,n for Nat;
reserve a,b,e for set;
reserve t,u,v,w,z for QC-symbol of A;
reserve p,q,r,s for Element of CQC-WFF(A);
reserve x for Element of bound_QC-variables(A);
reserve ll for CQC-variable_list of k,A;
reserve P for QC-pred_symbol of k,A;
reserve f,h for Element of Funcs(bound_QC-variables(A),bound_QC-variables(A)),
  K,L for Element of Fin bound_QC-variables(A);

theorem
  for k for l being CQC-variable_list of k,A for P being QC-pred_symbol of
  k,A holds SepQuadruples(P!l) = { [P!l,index(P!l),{}.bound_QC-variables(A),
  id bound_QC-variables(A)] }
proof
  let k;
  let l be CQC-variable_list of k,A;
  let P be QC-pred_symbol of k,A;
A1: P!l is atomic by QC_LANG1:def 18;
  now
    let x be object;
    thus x in SepQuadruples(P!l) implies x = [P!l,index(P!l),{}.
    bound_QC-variables(A),id bound_QC-variables(A)]
    proof
      assume
A2:   x in SepQuadruples(P!l);
      then consider q,t,K,f such that
A3:   x = [q,t,K,f] by DOMAIN_1:10;
A4:   now
        given x,u,h such that
        u++ = t and
        h +*({x} --> x.u) = f and
A5:     [All(x,q),u,K,h] in SepQuadruples(P!l) or [All(x,q),u,K\{.x
        .},h] in SepQuadruples(P!l);
        All(x,q) is_subformula_of P!l by A5,Th35;
        then All(x,q) = P!l by QC_LANG2:80;
        then P!l is universal by QC_LANG1:def 21;
        hence contradiction by A1,QC_LANG1:20;
      end;
A6:   now
        given r,u such that
        t = u+QuantNbr r and
A7:     [r '&' q,u,K,f] in SepQuadruples(P!l);
        r '&' q is_subformula_of P!l by A7,Th35;
        then r '&' q = P!l by QC_LANG2:80;
        then P!l is conjunctive by QC_LANG1:def 20;
        hence contradiction by A1,QC_LANG1:20;
      end;
A8:   now
        given r such that
A9:     [q '&' r, t, K,f] in SepQuadruples(P!l);
        q '&' r is_subformula_of P!l by A9,Th35;
        then q '&' r = P!l by QC_LANG2:80;
        then P!l is conjunctive by QC_LANG1:def 20;
        hence contradiction by A1,QC_LANG1:20;
      end;
A10:  now
        assume ['not' q,t,K,f] in SepQuadruples(P!l);
        then 'not' q is_subformula_of P!l by Th35;
        then 'not' q = P!l by QC_LANG2:80;
        then P!l is negative by QC_LANG1:def 19;
        hence contradiction by A1,QC_LANG1:20;
      end;
     set p = P!l;
  [q,t,K,f] = [p,index p,{}.
bound_QC-variables(A),id bound_QC-variables(A)] or
   ['not' q,t,K,f]
in SepQuadruples p or
   (ex r st [q '&' r, t, K,f] in SepQuadruples p) or
  (ex r,u st t = u+QuantNbr r & [r '&' q,u,K,f] in SepQuadruples p) or
  ex x,u,h st u++ = t & h +*({x} --> x.u) = f & ([All(x,q),u,K,h]
  in SepQuadruples p or
     [All(x,q),u,K\{x},h] in SepQuadruples p) by A2,Th34,A3;
      hence thesis by A3,A8,A6,A4,A10;
    end;
    thus x = [P!l,index(P!l),{}.bound_QC-variables(A),id bound_QC-variables(A)]
    implies x in SepQuadruples(P!l) by Th30;
  end;
  hence thesis by TARSKI:def 1;
end;
