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

theorem Th46:
  ( p => q ) => (( s '&' p ) => ( s '&' q )) in HP_TAUT
proof
  set P = p => q, Q = p '&' s, S = s '&' q, T = s '&' p;
A1: P => (Q => S) in HP_TAUT by Th45;
A2: T => Q in HP_TAUT by Th43;
  (P => (Q => S)) => ((T => Q) => (P => (T => S))) in HP_TAUT by Th24;
  then (T => Q) => (P => (T => S)) in HP_TAUT by A1,Def10;
  hence thesis by A2,Def10;
end;
