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 Th55:
  p <==> q & r <==> s implies p => r <==> q => s
proof
  assume that
A1: p <==> q and
A2: r <==> s;
  'not' r <==> 'not' s by A2,Lm5;
  then p '&' 'not' r <==> q '&' 'not' s by A1,Th54;
  then
A3: 'not' (p '&' 'not' r) <==> 'not' (q '&' 'not' s) by Lm5;
  p => r = 'not' (p '&' 'not' r) by QC_LANG2:def 2;
  hence thesis by A3,QC_LANG2:def 2;
end;
