set f = cos ^ ;
].(PI / 2),PI .] /\ (cos " {0 }) = {}

proof

assume ].(PI / 2),PI .] /\ (cos " {0 }) <> {} ; :: thesis: contradiction
then consider rr being set such that
A1: rr in ].(PI / 2),PI .] /\ (cos " {0 }) by XBOOLE_0:def 1;
rr in cos " {0 } by A1, XBOOLE_0:def 4;
then A2: cos . rr in {0 } by FUNCT_1:def 13;
rr in ].(PI / 2),PI .] by A1, XBOOLE_0:def 4;
then cos . rr <> 0 by Lm2, COMPTRIG:29;
hence contradiction by A2, TARSKI:def 1; :: thesis: verum

end;

then ( ].(PI / 2),PI .] \ (cos " {0 }) c= (dom cos ) \ (cos " {0 }) & ].(PI / 2),PI .] misses cos " {0 } ) by SIN_COS:27, XBOOLE_0:def 7, XBOOLE_1:33;
then ].(PI / 2),PI .] c= (dom cos ) \ (cos " {0 }) by XBOOLE_1:83;
hence ].(PI / 2),PI .] c= dom sec by RFUNCT_1:def 8; :: thesis: verum