let h, x be Real; :: thesis: ( x + (h / 2) in dom cot & x - (h / 2) in dom cot implies (cD (((cot (#) cot) (#) sin),h)) . x = ((cot (#) cos) . (x + (h / 2))) - ((cot (#) cos) . (x - (h / 2))) )
set f = (cot (#) cot) (#) sin;
assume A1: ( x + (h / 2) in dom cot & x - (h / 2) in dom cot ) ; :: thesis: (cD (((cot (#) cot) (#) sin),h)) . x = ((cot (#) cos) . (x + (h / 2))) - ((cot (#) cos) . (x - (h / 2)))
( x + (h / 2) in dom ((cot (#) cot) (#) sin) & x - (h / 2) in dom ((cot (#) cot) (#) sin) )

proof

set f1 = cot (#) cot;
set f2 = sin ;
A2: ( x + (h / 2) in dom (cot (#) cot) & x - (h / 2) in dom (cot (#) cot) )

proof

( x + (h / 2) in (dom cot) /\ (dom cot) & x - (h / 2) in (dom cot) /\ (dom cot) ) by A1;
hence ( x + (h / 2) in dom (cot (#) cot) & x - (h / 2) in dom (cot (#) cot) ) by VALUED_1:def 4; :: thesis: verum

end;

( x + (h / 2) in (dom (cot (#) cot)) /\ (dom sin) & x - (h / 2) in (dom (cot (#) cot)) /\ (dom sin) ) by A2, SIN_COS:24, XBOOLE_0:def 4;
hence ( x + (h / 2) in dom ((cot (#) cot) (#) sin) & x - (h / 2) in dom ((cot (#) cot) (#) sin) ) by VALUED_1:def 4; :: thesis: verum

end;

then (cD (((cot (#) cot) (#) sin),h)) . x = (((cot (#) cot) (#) sin) . (x + (h / 2))) - (((cot (#) cot) (#) sin) . (x - (h / 2))) by DIFF_1:39
.= (((cot (#) cot) . (x + (h / 2))) * (sin . (x + (h / 2)))) - (((cot (#) cot) (#) sin) . (x - (h / 2))) by VALUED_1:5
.= (((cot . (x + (h / 2))) * (cot . (x + (h / 2)))) * (sin . (x + (h / 2)))) - (((cot (#) cot) (#) sin) . (x - (h / 2))) by VALUED_1:5
.= (((cot . (x + (h / 2))) * (cot . (x + (h / 2)))) * (sin . (x + (h / 2)))) - (((cot (#) cot) . (x - (h / 2))) * (sin . (x - (h / 2)))) by VALUED_1:5
.= (((cot . (x + (h / 2))) * (cot . (x + (h / 2)))) * (sin . (x + (h / 2)))) - (((cot . (x - (h / 2))) * (cot . (x - (h / 2)))) * (sin . (x - (h / 2)))) by VALUED_1:5
.= ((((cos . (x + (h / 2))) * ((sin . (x + (h / 2))) ")) * (cot . (x + (h / 2)))) * (sin . (x + (h / 2)))) - (((cot . (x - (h / 2))) * (cot . (x - (h / 2)))) * (sin . (x - (h / 2)))) by A1, RFUNCT_1:def 1
.= ((((cos . (x + (h / 2))) * ((sin . (x + (h / 2))) ")) * (cot . (x + (h / 2)))) * (sin . (x + (h / 2)))) - ((((cos . (x - (h / 2))) * ((sin . (x - (h / 2))) ")) * (cot . (x - (h / 2)))) * (sin . (x - (h / 2)))) by A1, RFUNCT_1:def 1
.= (((cot . (x + (h / 2))) * (cos . (x + (h / 2)))) * ((sin . (x + (h / 2))) * (1 / (sin . (x + (h / 2)))))) - (((cot . (x - (h / 2))) * (cos . (x - (h / 2)))) * ((sin . (x - (h / 2))) * (1 / (sin . (x - (h / 2))))))
.= (((cot . (x + (h / 2))) * (cos . (x + (h / 2)))) * 1) - (((cot . (x - (h / 2))) * (cos . (x - (h / 2)))) * ((sin . (x - (h / 2))) * (1 / (sin . (x - (h / 2)))))) by A1, FDIFF_8:2, XCMPLX_1:106
.= (((cot . (x + (h / 2))) * (cos . (x + (h / 2)))) * 1) - (((cot . (x - (h / 2))) * (cos . (x - (h / 2)))) * 1) by A1, FDIFF_8:2, XCMPLX_1:106
.= ((cot (#) cos) . (x + (h / 2))) - ((cot . (x - (h / 2))) * (cos . (x - (h / 2)))) by VALUED_1:5
.= ((cot (#) cos) . (x + (h / 2))) - ((cot (#) cos) . (x - (h / 2))) by VALUED_1:5 ;
hence (cD (((cot (#) cot) (#) sin),h)) . x = ((cot (#) cos) . (x + (h / 2))) - ((cot (#) cos) . (x - (h / 2))) ; :: thesis: verum