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

theorem Th31:
  (p '&' q) in HP_TAUT iff p in HP_TAUT & q in HP_TAUT
proof
  hereby
A1: (p '&' q) => q in HP_TAUT by Def10;
    assume
A2: p '&' q in HP_TAUT;
    (p '&' q) => p in HP_TAUT by Def10;
    hence p in HP_TAUT & q in HP_TAUT by A2,A1,Def10;
  end;
  assume that
A3: p in HP_TAUT and
A4: q in HP_TAUT;
  p => (q => (p '&' q)) in HP_TAUT by Def10;
  then q => (p '&' q) in HP_TAUT by A3,Def10;
  hence thesis by A4,Def10;
end;
