let x be Real; :: thesis: ( x in dom tan implies cos . x <> 0 )
assume x in dom tan ; :: thesis: cos . x <> 0
then x in (dom sin ) /\ ((dom cos ) \ (cos " {0 })) by RFUNCT_1:def 4;
then x in (dom cos ) \ (cos " {0 }) by XBOOLE_0:def 4;
then x in dom (cos ^ ) by RFUNCT_1:def 8;
hence cos . x <> 0 by RFUNCT_1:13; :: thesis: verum