reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,j,k,m,n for Nat,
  p,q,r for Element of CQC-WFF(Al),
  x,y,y0 for bound_QC-variable of Al,
  X for Subset of CQC-WFF(Al),
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  Sub for CQC_Substitution of Al,
  f,f1,g,h,h1 for FinSequence of CQC-WFF(Al);
reserve fin,fin1 for FinSequence;
reserve PR,PR1 for FinSequence of [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:];
reserve a for Element of A;

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;
