reserve Al for QC-alphabet;
reserve i,j,n,k,l for Nat;
reserve a for set;
reserve T,S,X,Y for Subset of CQC-WFF(Al);
reserve p,q,r,t,F,H,G for Element of CQC-WFF(Al);
reserve s for QC-formula of Al;
reserve x,y for bound_QC-variable of Al;

theorem
  T is being_a_theory & S is being_a_theory implies T /\ S is being_a_theory
proof
  assume that
A1: T is being_a_theory and
A2: S is being_a_theory;
 VERUM(Al) in T & VERUM(Al) in S by A1,A2;
  hence VERUM(Al) in T /\ S by XBOOLE_0:def 4;
  let p,q,r,s,x,y;
   ('not' p => p) => p in T & ('not' p => p) => p in S by A1,A2;
  hence ('not' p => p) => p in T /\ S by XBOOLE_0:def 4;
   p => ('not' p => q) in T & p => ('not' p => q) in S by A1,A2;
  hence p => ('not' p => q) in T /\ S by XBOOLE_0:def 4;
   (
p => q) => ('not'(q '&' r) => 'not'(p '&' r)) in T & (p => q) => ('not'(q
  '&' r) => 'not'(p '&' r)) in S by A1,A2;
  hence (p => q) => ('not'(q '&' r) => 'not'
  (p '&' r)) in T /\ S by XBOOLE_0:def 4;
   p '&' q => q '&' p in T & p '&' q => q '&' p in S by A1,A2;
  hence p '&' q => q '&' p in T /\ S by XBOOLE_0:def 4;
A3: p in T & p => q in T implies q in T by A1;
 p in S & p => q in S implies q in S by A2;
  hence p in T /\ S & p => q in T /\ S implies q in T /\ S by A3,XBOOLE_0:def 4
;
 All(x,p) => p in T & All(x,p) => p in S by A1,A2;
  hence All(x,p) => p in T /\ S by XBOOLE_0:def 4;
A4: p => q in T & not x in still_not-bound_in p implies p => All(x,q) in T
  by A1;
 p => q in S & not x in still_not-bound_in p implies p => All(x,q) in S
  by A2;
  hence p => q in T /\ S & not x in still_not-bound_in p implies
  p => All(x,q) in T /\ S by A4,XBOOLE_0:def 4;
A5: s.x in CQC-WFF(Al) & s.y in CQC-WFF(Al) & not x in still_not-bound_in s &
s.x in T implies s.y in T by A1;
 s.x in CQC-WFF(Al) & s.y in CQC-WFF(Al) & not x in still_not-bound_in s & s.x
  in S implies s.y in S by A2;
  hence thesis by A5,XBOOLE_0:def 4;
end;
