set f = sin ^ ;
A1: ].0 ,(PI / 2).] \ (sin " {0 }) c= (dom sin ) \ (sin " {0 }) by SIN_COS:27, XBOOLE_1:33;
].0 ,(PI / 2).] /\ (sin " {0 }) = {}

proof

assume ].0 ,(PI / 2).] /\ (sin " {0 }) <> {} ; :: thesis: contradiction
then consider rr being set such that
A2: rr in ].0 ,(PI / 2).] /\ (sin " {0 }) by XBOOLE_0:def 1;
A3: ( rr in ].0 ,(PI / 2).] & rr in sin " {0 } ) by A2, XBOOLE_0:def 4;
A5: sin . rr <> 0 by A3, Lm4, COMPTRIG:23;
sin . rr in {0 } by A3, FUNCT_1:def 13;
hence contradiction by A5, TARSKI:def 1; :: thesis: verum

end;

then ].0 ,(PI / 2).] misses sin " {0 } by XBOOLE_0:def 7;
then ].0 ,(PI / 2).] c= (dom sin ) \ (sin " {0 }) by A1, XBOOLE_1:83;
hence ].0 ,(PI / 2).] c= dom cosec by RFUNCT_1:def 8; :: thesis: verum