
theorem
  for x,y,z being Element of BOOLEAN holds
   and3a.<*x,y,z*> = nor3b.<*z,y,x*> & and3b.<*x,y,z*> = nor3a.<*z,y,x*>
proof
  let x,y,z be Element of BOOLEAN;
  thus and3a.<*x,y,z*> = 'not' ('not' 'not' ('not' 'not' x 'or' 'not' y) 'or'
  'not' z) by Def10
    .= 'not' (('not' z 'or' 'not' y) 'or' x)
    .= nor3b.<*z,y,x*> by Def19;
  thus and3b.<*x,y,z*> = 'not' ('not' 'not' ('not' 'not' x 'or' 'not' 'not' y)
  'or' 'not' z) by Def11
    .= 'not' (('not' z 'or' y) 'or' x)
    .= nor3a.<*z,y,x*> by Def18;
end;
