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

proof

assume ].0 ,(PI / 2).] /\ (sin " {0 }) <> {} ; :: thesis: contradiction
then consider rr being set such that
A1: rr in ].0 ,(PI / 2).] /\ (sin " {0 }) by XBOOLE_0:def 1;
rr in sin " {0 } by A1, XBOOLE_0:def 4;
then A2: sin . rr in {0 } by FUNCT_1:def 13;
rr in ].0 ,(PI / 2).] by A1, XBOOLE_0:def 4;
then sin . rr <> 0 by Lm4, COMPTRIG:23;
hence contradiction by A2, TARSKI:def 1; :: thesis: verum

end;

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