let A be symmetrical Subset of COMPLEX ; :: thesis: ( A c= dom tan & ( for x being Real st x in A holds
cos . x <> 0 ) implies tan is_odd_on A )
assume that
A1: A c= dom tan and
A2: for x being Real st x in A holds
cos . x <> 0 ; :: thesis: tan is_odd_on A
B1: for x being Real st x in A holds
tan . (- x) = - (tan . x)

proof

let x be Real; :: thesis: ( x in A implies tan . (- x) = - (tan . x) )
assume A4: x in A ; :: thesis: tan . (- x) = - (tan . x)
then - x in A by Def1;
then tan . (- x) = tan (- x) by A2, SIN_COS9:15
.= - (tan x) by SIN_COS4:2
.= - (tan . x) by A2, A4, SIN_COS9:15 ;
hence tan . (- x) = - (tan . x) ; :: thesis: verum

end;

B2: dom (tan | A) = A by RELAT_1:91, A1;
for x being Real st x in dom (tan | A) & - x in dom (tan | A) holds
(tan | A) . (- x) = - ((tan | A) . x)

proof

let x be Real; :: thesis: ( x in dom (tan | A) & - x in dom (tan | A) implies (tan | A) . (- x) = - ((tan | A) . x) )
assume A3: ( x in dom (tan | A) & - x in dom (tan | A) ) ; :: thesis: (tan | A) . (- x) = - ((tan | A) . x)
(tan | A) . (- x) = (tan | A) /. (- x) by PARTFUN1:def 8, A3
.= tan /. (- x) by PARTFUN2:35, B2, A3, A1
.= tan . (- x) by PARTFUN1:def 8, A1, B2, A3
.= - (tan . x) by B1, B2, A3
.= - (tan /. x) by PARTFUN1:def 8, A1, B2, A3
.= - ((tan | A) /. x) by PARTFUN2:35, B2, A3, A1
.= - ((tan | A) . x) by PARTFUN1:def 8, A3 ;
hence (tan | A) . (- x) = - ((tan | A) . x) ; :: thesis: verum

end;

then ( tan | A is with_symmetrical_domain & tan | A is quasi_odd ) by Def6, B2, Def2;
hence tan is_odd_on A by A1, Def8; :: thesis: verum