reserve T, X, Y for Subset of HP-WFF;
reserve p, q, r, s for Element of HP-WFF;

theorem Th34:
  ( p => ( q => r )) => (( p '&' q ) => r ) in HP_TAUT
proof
A1: (( p '&' q ) => q) => (( q => r ) => (( p '&' q ) => r )) in HP_TAUT by
Th21;
  ( p '&' q ) => q in HP_TAUT by Def10;
  then ( q => r ) => (( p '&' q ) => r ) in HP_TAUT by A1,Def10;
  then
A2: p => (( q => r ) => (( p '&' q ) => r )) in HP_TAUT by Th15;
A3: ( p => (( p '&' q ) => r )) => ((p '&' q ) => ( p => r )) in HP_TAUT by
Th26;
  p => (( q => r ) => (( p '&' q ) => r )) => ((p => ( q => r )) => ( p =>
  (( p '&' q ) => r ))) in HP_TAUT by Def10;
  then (p => ( q => r )) => ( p => (( p '&' q ) => r )) in HP_TAUT by A2,Def10;
  then
A4: (p => ( q => r )) => ((p '&' q ) => ( p => r )) in HP_TAUT by A3,Th23;
A5: ( p '&' q ) => p in HP_TAUT by Def10;
  ((p '&' q ) => ( p => r )) => ((( p '&' q ) => p ) => (( p '&' q ) => r
  )) in HP_TAUT by Def10;
  then ((p '&' q ) => ( p => r )) => (( p '&' q ) => r ) in HP_TAUT by A5,Th29;
  hence thesis by A4,Th23;
end;
