
theorem Th17:
  <*FALSE*> + <*TRUE*> = <*TRUE*> & <*TRUE*> + <*FALSE*> = <*TRUE *>
proof
  add_ovfl(<*FALSE*>,<*TRUE*>) <> TRUE by Th13;
  then add_ovfl(<*FALSE*>,<*TRUE*>) = FALSE by XBOOLEAN:def 3;
  then <*FALSE*>,<*TRUE*> are_summable by BINARITH:def 7;
  then Absval(<*FALSE*> + <*TRUE*>) = Absval <*FALSE*> + Absval <*TRUE*> by
BINARITH:22
    .= Absval <*FALSE*> + 1 by BINARITH:16
    .= 0 + 1 by BINARITH:15
    .= Absval <*TRUE*> by BINARITH:16;
  hence <*FALSE*> + <*TRUE*> = <*TRUE*> by Th2;
  add_ovfl(<*TRUE*>,<*FALSE*>) <> TRUE by Th13;
  then add_ovfl(<*TRUE*>,<*FALSE*>) = FALSE by XBOOLEAN:def 3;
  then <*TRUE*>,<*FALSE*> are_summable by BINARITH:def 7;
  then Absval(<*TRUE*> + <*FALSE*>) = Absval <*TRUE*> + Absval <*FALSE*> by
BINARITH:22
    .= Absval <*TRUE*> + 0 by BINARITH:15
    .= Absval <*TRUE*>;
  hence thesis by Th2;
end;
