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 Th36:
  All(x,p '&' q) => (All(x,p) '&' All(x,q)) is valid & (All(x,p)
  '&' All(x,q)) => All(x,p '&' q) is valid
proof
  All(x,p) => p is valid & All(x,q) => q is valid by CQC_THE1:66;
  then
A1: (All(x,p) '&' All(x,q)) => (p '&' q) is valid by Lm5;
  All(x,p '&'q => q) is valid & All(x,p '&' q => q) => (All(x,p '&' q) =>
  All( x,q)) is valid by Lm1,Th23,Th30;
  then
A2: All(x,p '&' q) => All(x,q) is valid by CQC_THE1:65;
  All(x,p '&' q => p) is valid & All(x,p '&' q => p) => (All(x,p '&' q) =>
  All (x,p)) is valid by Lm1,Th23,Th30;
  then All(x,p '&' q) => All(x,p) is valid by CQC_THE1:65;
  hence All(x,p '&' q) => (All(x,p) '&' All(x,q)) is valid by A2,Lm3;
  ( not x in still_not-bound_in All(x,p))& not x in still_not-bound_in
  All( x, q ) by Th5;
  then not x in still_not-bound_in All(x,p) '&' All(x,q) by Th8;
  hence thesis by A1,CQC_THE1:67;
end;
