reserve A for QC-alphabet;
reserve p, q, r, s for Element of CQC-WFF(A);

theorem Th51:
  p => q in TAUT(A) implies ( p '&' r ) => ( q '&' r ) in TAUT(A)
proof
A1: ( p => q ) => (( q => 'not' r ) => ( p => 'not' r )) in TAUT(A)
by LUKASI_1:1;
  assume p => q in TAUT(A);
  then ( q => 'not' r ) => ( p => 'not' r ) in TAUT(A) by A1,CQC_THE1:46;
  then
A2: 'not' ( p => 'not' r ) => 'not' ( q => 'not' r ) in TAUT(A) by LUKASI_1:34;
A3: 'not' ( q => 'not' r ) => ( q '&' r ) in TAUT(A) by Th16;
  ( p '&' r ) => 'not' ( p => 'not' r ) in TAUT(A) by Th15;
  then ( p '&' r ) => 'not' ( q => 'not' r ) in TAUT(A) by A2,LUKASI_1:3;
  hence thesis by A3,LUKASI_1:3;
end;
