reserve Al for QC-alphabet;
reserve p,q,p1,p2,q1 for Element of CQC-WFF(Al),
  k for Element of NAT,
  f,f1,f2,g for FinSequence of CQC-WFF(Al),
  a,b,b1,b2,c,i,n for Nat;
reserve P for Permutation of dom f;

theorem Th27:
  |- f^<*p*>^<*q*> implies |- f^<*p => q*>
proof
  assume
A1: |- f^<*p*>^<*q*>;
  set g1 = f^<*p '&' 'not' q*>^<*p*>^<*q*>;
  set g = f^<*p*>^<*q*>;
A2: Ant(g1) = f^<*p '&' 'not' q*>^<*p*> by CALCUL_1:5;
  Suc(g) = q by CALCUL_1:5;
  then
A3: Suc(g1) = Suc(g) by CALCUL_1:5;
A4: Ant(g) = f^<*p*> by CALCUL_1:5;
  then
A5: 0+1 <= len Ant(g) by CALCUL_1:10;
A6: |- f^<*p '&' 'not' q*>^<*p '&' 'not' q*> by Th21;
  then
A7: |- f^<*p '&' 'not' q*>^<*p*> by Th22;
  Ant(Ant(g)) = f & Suc(Ant(g)) = p by A4,CALCUL_1:5;
  then |- g1 by A1,A5,A3,A2,CALCUL_1:13,36;
  then
A8: |- f^<*p '&' 'not' q*>^<*q*> by A7,Th24;
A9: |- f^<*'not' (p '&' 'not' q)*>^<*'not' (p '&' 'not' q)*> by Th21;
  |- f^<*p '&' 'not' q*>^<*'not' q*> by A6,Th23;
  then |- f^<*p '&' 'not' q*>^<*'not' (p '&' 'not' q)*> by A8,Th25;
  then |- f^<*'not' (p '&' 'not' q)*> by A9,Th26;
  hence thesis by QC_LANG2:def 2;
end;
