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 Th39:
  (All(x,p) 'or' All(x,q)) => All(x,p 'or' q) is valid
proof
  All(x,q => p 'or' q) is valid & All(x,q => p 'or' q) => (All(x,q) => All
  (x,p 'or' q)) is valid by Lm6,Th23,Th30;
  then
A1: All(x,q) => All(x,p 'or' q) is valid by CQC_THE1:65;
  All(x,p => p 'or' q) is valid & All(x,p => p 'or' q) => (All(x,p) => All
  (x,p 'or' q)) is valid by Lm6,Th23,Th30;
  then All(x,p) => All(x,p 'or' q) is valid by CQC_THE1:65;
  hence thesis by A1,Lm7;
end;
