reserve Al for QC-alphabet;
reserve b,c,d for set,
  X,Y for Subset of CQC-WFF(Al),
  i,j,k,m,n for Nat,
  p,p1,q,r,s,s1 for Element of CQC-WFF(Al),
  x,x1,x2,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al, A,
  v for Element of Valuations_in(Al,A),
  f1,f2 for FinSequence of CQC-WFF(Al),
  CX,CY,CZ for Consistent Subset of CQC-WFF(Al),
  JH for Henkin_interpretation of CX,
  a for Element of A,
  t,u for QC-symbol of Al;

theorem Th5:
  |- f1^<*p*> & |- f1^<*q*> implies |- f1^<*p '&' q*>
proof
  set g = f1^<*p*>;
  set g1 = f1^<*q*>;
  assume that
A1: |- g and
A2: |- g1;
A3: Ant(g) = f1 by CALCUL_1:5;
A4: Suc(g) = p by CALCUL_1:5;
A5: Suc(g1) = q by CALCUL_1:5;
  Ant(g) = Ant(g1) by A3,CALCUL_1:5;
  hence thesis by A1,A2,A3,A4,A5,CALCUL_1:39;
end;
