theorem Th8:
  for p,r,x,P,l for Al2 being Al-expanding QC-alphabet
  holds Al2-Cast(VERUM(Al)) = VERUM(Al2) &
  Al2-Cast(P!l) = Al2-Cast(P)!Al2-Cast(l) &
  Al2-Cast('not' p) = 'not' (Al2-Cast(p)) &
  Al2-Cast(p '&' r) = (Al2-Cast(p)) '&' (Al2-Cast(r)) &
  Al2-Cast(All(x,p)) = All(Al2-Cast(x),Al2-Cast(p))
proof
  let p,r,x,P,l;
  let Al2 be Al-expanding QC-alphabet;
A1: the_arity_of P = len l by Th1;
A2: the_arity_of Al2-Cast(P) = len (Al2-Cast(l)) by Th1;
  thus Al2-Cast(VERUM(Al)) = VERUM(Al2);
  thus Al2-Cast(P!l) = <*P*>^l by A1,QC_LANG1:def 12
           .= Al2-Cast(P)!Al2-Cast(l) by A2,QC_LANG1:def 12;
  thus Al2-Cast('not' p) = 'not' (Al2-Cast(p));
  thus Al2-Cast(p '&' r) = (Al2-Cast(p)) '&' (Al2-Cast(r));
  thus Al2-Cast(All(x,p)) = All(Al2-Cast(x),Al2-Cast(p));
end;
