let h, x be Real; :: thesis: ( x in dom tan & x + h in dom tan implies (fD (((tan (#) tan) (#) cos),h)) . x = ((tan (#) sin) . (x + h)) - ((tan (#) sin) . x) )
set f = (tan (#) tan) (#) cos;
assume A1: ( x in dom tan & x + h in dom tan ) ; :: thesis: (fD (((tan (#) tan) (#) cos),h)) . x = ((tan (#) sin) . (x + h)) - ((tan (#) sin) . x)
( x in dom ((tan (#) tan) (#) cos) & x + h in dom ((tan (#) tan) (#) cos) )

proof

set f1 = tan (#) tan;
set f2 = cos ;
A2: ( x in dom (tan (#) tan) & x + h in dom (tan (#) tan) )

proof

( x in (dom tan) /\ (dom tan) & x + h in (dom tan) /\ (dom tan) ) by A1;
hence ( x in dom (tan (#) tan) & x + h in dom (tan (#) tan) ) by VALUED_1:def 4; :: thesis: verum

end;

( x in (dom (tan (#) tan)) /\ (dom cos) & x + h in (dom (tan (#) tan)) /\ (dom cos) ) by A2, SIN_COS:24, XBOOLE_0:def 4;
hence ( x in dom ((tan (#) tan) (#) cos) & x + h in dom ((tan (#) tan) (#) cos) ) by VALUED_1:def 4; :: thesis: verum

end;

then (fD (((tan (#) tan) (#) cos),h)) . x = (((tan (#) tan) (#) cos) . (x + h)) - (((tan (#) tan) (#) cos) . x) by DIFF_1:1
.= (((tan (#) tan) . (x + h)) * (cos . (x + h))) - (((tan (#) tan) (#) cos) . x) by VALUED_1:5
.= (((tan . (x + h)) * (tan . (x + h))) * (cos . (x + h))) - (((tan (#) tan) (#) cos) . x) by VALUED_1:5
.= (((tan . (x + h)) * (tan . (x + h))) * (cos . (x + h))) - (((tan (#) tan) . x) * (cos . x)) by VALUED_1:5
.= (((tan . (x + h)) * (tan . (x + h))) * (cos . (x + h))) - (((tan . x) * (tan . x)) * (cos . x)) by VALUED_1:5
.= ((((sin . (x + h)) * ((cos . (x + h)) ")) * (tan . (x + h))) * (cos . (x + h))) - (((tan . x) * (tan . x)) * (cos . x)) by A1, RFUNCT_1:def 1
.= ((((sin . (x + h)) * ((cos . (x + h)) ")) * (tan . (x + h))) * (cos . (x + h))) - ((((sin . x) * ((cos . x) ")) * (tan . x)) * (cos . x)) by A1, RFUNCT_1:def 1
.= (((sin . (x + h)) * (tan . (x + h))) * ((cos . (x + h)) * (1 / (cos . (x + h))))) - (((sin . x) * (tan . x)) * ((cos . x) * (1 / (cos . x))))
.= (((sin . (x + h)) * (tan . (x + h))) * 1) - (((sin . x) * (tan . x)) * ((cos . x) * (1 / (cos . x)))) by A1, FDIFF_8:1, XCMPLX_1:106
.= (((sin . (x + h)) * (tan . (x + h))) * 1) - (((sin . x) * (tan . x)) * 1) by A1, FDIFF_8:1, XCMPLX_1:106
.= ((tan (#) sin) . (x + h)) - ((tan . x) * (sin . x)) by VALUED_1:5
.= ((tan (#) sin) . (x + h)) - ((tan (#) sin) . x) by VALUED_1:5 ;
hence (fD (((tan (#) tan) (#) cos),h)) . x = ((tan (#) sin) . (x + h)) - ((tan (#) sin) . x) ; :: thesis: verum