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 Th6:
  X |- p & X |- q iff X |- p '&' q
proof
  thus X |- p & X |- q implies X |- p '&' q
  proof
    assume that
A1: X |- p and
A2: X |- q;
    consider f1 such that
A3: rng f1 c= X and
A4: |- f1^<*p*> by A1,HENMODEL:def 1;
    consider f2 such that
A5: rng f2 c= X and
A6: |- f2^<*q*> by A2,HENMODEL:def 1;
A7: |- f1^f2^<*p*> by A4,HENMODEL:5;
    |- f1^f2^<*q*> by A6,CALCUL_2:20;
    then
A8: |- f1^f2^<*p '&' q*> by A7,Th5;
    rng(f1^f2) = rng f1 \/ rng f2 by FINSEQ_1:31;
    then rng(f1^f2) c= X by A3,A5,XBOOLE_1:8;
    hence thesis by A8,HENMODEL:def 1;
  end;
  thus X |- p '&' q implies X |- p & X |- q
  proof
    assume X |- p '&' q;
    then consider f1 such that
A9: rng f1 c= X and
A10: |- f1^<*p '&' q*> by HENMODEL:def 1;
A11: |- f1^<*p*> by A10,CALCUL_2:22;
    |- f1^<*q*> by A10,CALCUL_2:23;
    hence thesis by A9,A11,HENMODEL:def 1;
  end;
end;
