reserve A for QC-alphabet;
reserve p, q, r, s, p1, q1 for Element of CQC-WFF(A),
  X, Y, Z, X1, X2 for Subset of CQC-WFF(A),
  h for QC-formula of A,
  x, y for bound_QC-variable of A,
  n for Element of NAT;

theorem Th9:
  X |- Y & Y |- Z implies X |- Z
proof
  assume that
A1: X |- Y and
A2: Y |- Z;
  for p st p in Z holds X |- p
  proof
    let p;
    assume p in Z;
    then Y |- p by A2;
    then
A3: p in Cn(Y) by CQC_THE1:def 8;
    Y c= Cn(X) by A1,Th7;
    then Cn(Y) c= Cn(X) by CQC_THE1:15,16;
    hence thesis by A3,CQC_THE1:def 8;
  end;
  hence thesis;
end;
