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
  p <==> q & r <==> s implies p 'or' r <==> q 'or' s
proof
  assume that
A1: p <==> q and
A2: r <==> s;
  s => r is valid by A2,Th50;
  then
A3: s => r in TAUT(A) by CQC_THE1:def 10;
  q => p is valid by A1,Th50;
  then q => p in TAUT(A) by CQC_THE1:def 10;
  then q 'or' s => p 'or' r in TAUT(A) by A3,PROCAL_1:57;
  then
A4: q 'or' s => p 'or' r is valid by CQC_THE1:def 10;
  r => s is valid by A2,Th50;
  then
A5: r => s in TAUT(A) by CQC_THE1:def 10;
  p => q is valid by A1,Th50;
  then p => q in TAUT(A) by CQC_THE1:def 10;
  then p 'or' r => q 'or' s in TAUT(A) by A5,PROCAL_1:57;
  then p 'or' r => q 'or' s is valid by CQC_THE1:def 10;
  hence thesis by A4,Th50;
end;
