A1: for x being Element of REAL st x in dom (cos `| REAL) holds
(cos `| REAL) . x = (- sin) . x

proof

let x be Element of REAL ; :: thesis: ( x in dom (cos `| REAL) implies (cos `| REAL) . x = (- sin) . x )
assume x in dom (cos `| REAL) ; :: thesis: (cos `| REAL) . x = (- sin) . x
(cos `| REAL) . x = diff (cos,x) by FDIFF_1:def 7, SIN_COS:67
.= - (sin . x) by SIN_COS:63 ;
hence (cos `| REAL) . x = (- sin) . x by VALUED_1:8; :: thesis: verum

end;

dom (- sin) = REAL by FUNCT_2:def 1;
then dom (cos `| REAL) = dom (- sin) by FDIFF_1:def 7, SIN_COS:67;
hence cos `| REAL = - sin by A1, PARTFUN1:5; :: thesis: verum