reserve A for QC-alphabet;
reserve X,T for Subset of CQC-WFF(A);
reserve F,G,H,p,q,r,t for Element of CQC-WFF(A);
reserve s,h for QC-formula of A;
reserve x,y for bound_QC-variable of A;
reserve f for FinSequence of [:CQC-WFF(A),Proof_Step_Kinds:];
reserve i,j for Element of NAT;

theorem Th65:
  All(x,p '&' q) => (All(x,p) '&' q) is valid
proof
A1: (q '&' All(x,p)) => (All(x,p) '&' q) is valid by CQC_THE1:64;
  All(x,(p '&' q) => (q '&' p)) is valid & All(x,(p '&' q) => (q '&' p))
  => ( All(x,p '&' q) => All(x,q '&' p)) is valid by Th23,Th30,CQC_THE1:64;
  then
A2: All(x,p '&' q) => All(x,q '&' p) is valid by CQC_THE1:65;
  All(x,q '&' p) => (q '&' All(x,p)) is valid by Th64;
  then All(x,p '&' q) => (q '&' All(x,p)) is valid by A2,LUKASI_1:42;
  hence thesis by A1,LUKASI_1:42;
end;
