theorem Th52:
  |- f^<*p*>^<*r*> & |- f^<*q*>^<*r*> implies |- f^<*p 'or' q*>^<* r*>
proof
  set f1 = f^<*'not' r*>^<*'not' p*>;
  set f2 = f^<*'not' r*>^<*'not' q*>;
A1: Ant(f1) = f^<*'not' r*> by Th5;
A2: Suc(f2) = 'not' q by Th5;
  assume |- f^<*p*>^<*r*> & |- f^<*q*>^<*r*>;
  then
A3: |- f^<*'not' r*>^<*'not' p*> & |- f^<*'not' r*>^<*'not' q*> by Th46;
  Suc(f1) = 'not' p & Ant(f2) = f^<*'not' r*> by Th5;
  then |- Ant(f1)^<*'not' p '&' 'not' q*> by A3,A2,Th5,Th39;
  then |- f^<*'not' ('not' p '&' 'not' q)*>^<*r*> by A1,Th48;
  hence thesis by QC_LANG2:def 3;
end;
