let A be symmetrical Subset of REAL; :: thesis: sinh is_odd_on A

A1: dom sinh = REAL by FUNCT_2:def 1;

then A2: dom (sinh | A) = A by RELAT_1:62;

for x being Real st x in dom (sinh | A) & - x in dom (sinh | A) holds

(sinh | A) . (- x) = - ((sinh | A) . x)

hence sinh is_odd_on A by A1; :: thesis: verum

A1: dom sinh = REAL by FUNCT_2:def 1;

then A2: dom (sinh | A) = A by RELAT_1:62;

for x being Real st x in dom (sinh | A) & - x in dom (sinh | A) holds

(sinh | A) . (- x) = - ((sinh | A) . x)

proof

then
( sinh | A is with_symmetrical_domain & sinh | A is quasi_odd )
by A2;
let x be Real; :: thesis: ( x in dom (sinh | A) & - x in dom (sinh | A) implies (sinh | A) . (- x) = - ((sinh | A) . x) )

assume that

A3: x in dom (sinh | A) and

A4: - x in dom (sinh | A) ; :: thesis: (sinh | A) . (- x) = - ((sinh | A) . x)

reconsider x = x as Element of REAL by XREAL_0:def 1;

(sinh | A) . (- x) = (sinh | A) /. (- x) by A4, PARTFUN1:def 6

.= sinh /. (- x) by A1, A4, PARTFUN2:17

.= - (sinh /. x) by SIN_COS2:19

.= - ((sinh | A) /. x) by A1, A3, PARTFUN2:17

.= - ((sinh | A) . x) by A3, PARTFUN1:def 6 ;

hence (sinh | A) . (- x) = - ((sinh | A) . x) ; :: thesis: verum

end;assume that

A3: x in dom (sinh | A) and

A4: - x in dom (sinh | A) ; :: thesis: (sinh | A) . (- x) = - ((sinh | A) . x)

reconsider x = x as Element of REAL by XREAL_0:def 1;

(sinh | A) . (- x) = (sinh | A) /. (- x) by A4, PARTFUN1:def 6

.= sinh /. (- x) by A1, A4, PARTFUN2:17

.= - (sinh /. x) by SIN_COS2:19

.= - ((sinh | A) /. x) by A1, A3, PARTFUN2:17

.= - ((sinh | A) . x) by A3, PARTFUN1:def 6 ;

hence (sinh | A) . (- x) = - ((sinh | A) . x) ; :: thesis: verum

hence sinh is_odd_on A by A1; :: thesis: verum