let h, x be Real; :: thesis: ( x + (h / 2) in dom tan & x - (h / 2) in dom tan implies (cD (((tan (#) tan) (#) sin),h)) . x = (((sin . (x + (h / 2))) |^ 3) * (((cos . (x + (h / 2))) ") |^ 2)) - (((sin . (x - (h / 2))) |^ 3) * (((cos . (x - (h / 2))) ") |^ 2)) )
set f = (tan (#) tan) (#) sin;
assume A1: ( x + (h / 2) in dom tan & x - (h / 2) in dom tan ) ; :: thesis: (cD (((tan (#) tan) (#) sin),h)) . x = (((sin . (x + (h / 2))) |^ 3) * (((cos . (x + (h / 2))) ") |^ 2)) - (((sin . (x - (h / 2))) |^ 3) * (((cos . (x - (h / 2))) ") |^ 2))
( x + (h / 2) in dom ((tan (#) tan) (#) sin) & x - (h / 2) in dom ((tan (#) tan) (#) sin) )

proof

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

proof

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

end;

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

end;

then (cD (((tan (#) tan) (#) sin),h)) . x = (((tan (#) tan) (#) sin) . (x + (h / 2))) - (((tan (#) tan) (#) sin) . (x - (h / 2))) by DIFF_1:39
.= (((tan (#) tan) . (x + (h / 2))) * (sin . (x + (h / 2)))) - (((tan (#) tan) (#) sin) . (x - (h / 2))) by VALUED_1:5
.= (((tan (#) tan) . (x + (h / 2))) * (sin . (x + (h / 2)))) - (((tan (#) tan) . (x - (h / 2))) * (sin . (x - (h / 2)))) by VALUED_1:5
.= (((tan . (x + (h / 2))) * (tan . (x + (h / 2)))) * (sin . (x + (h / 2)))) - (((tan (#) tan) . (x - (h / 2))) * (sin . (x - (h / 2)))) by VALUED_1:5
.= (((tan . (x + (h / 2))) * (tan . (x + (h / 2)))) * (sin . (x + (h / 2)))) - (((tan . (x - (h / 2))) * (tan . (x - (h / 2)))) * (sin . (x - (h / 2)))) by VALUED_1:5
.= ((((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) ")) * (tan . (x + (h / 2)))) * (sin . (x + (h / 2)))) - (((tan . (x - (h / 2))) * (tan . (x - (h / 2)))) * (sin . (x - (h / 2)))) by A1, RFUNCT_1:def 1
.= ((((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) ")) * ((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) "))) * (sin . (x + (h / 2)))) - (((tan . (x - (h / 2))) * (tan . (x - (h / 2)))) * (sin . (x - (h / 2)))) by A1, RFUNCT_1:def 1
.= ((((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) ")) * ((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) "))) * (sin . (x + (h / 2)))) - ((((sin . (x - (h / 2))) * ((cos . (x - (h / 2))) ")) * (tan . (x - (h / 2)))) * (sin . (x - (h / 2)))) by A1, RFUNCT_1:def 1
.= ((((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) ")) * ((sin . (x + (h / 2))) * ((cos . (x + (h / 2))) "))) * (sin . (x + (h / 2)))) - ((((sin . (x - (h / 2))) * ((cos . (x - (h / 2))) ")) * ((sin . (x - (h / 2))) * ((cos . (x - (h / 2))) "))) * (sin . (x - (h / 2)))) by A1, RFUNCT_1:def 1
.= ((((sin . (x + (h / 2))) * (sin . (x + (h / 2)))) * (sin . (x + (h / 2)))) * (((cos . (x + (h / 2))) ") * ((cos . (x + (h / 2))) "))) - ((((sin . (x - (h / 2))) * (sin . (x - (h / 2)))) * (sin . (x - (h / 2)))) * (((cos . (x - (h / 2))) ") * ((cos . (x - (h / 2))) ")))
.= (((((sin . (x + (h / 2))) |^ 1) * (sin . (x + (h / 2)))) * (sin . (x + (h / 2)))) * (((cos . (x + (h / 2))) ") * ((cos . (x + (h / 2))) "))) - ((((sin . (x - (h / 2))) * (sin . (x - (h / 2)))) * (sin . (x - (h / 2)))) * (((cos . (x - (h / 2))) ") * ((cos . (x - (h / 2))) ")))
.= ((((sin . (x + (h / 2))) |^ (1 + 1)) * (sin . (x + (h / 2)))) * (((cos . (x + (h / 2))) ") * ((cos . (x + (h / 2))) "))) - ((((sin . (x - (h / 2))) * (sin . (x - (h / 2)))) * (sin . (x - (h / 2)))) * (((cos . (x - (h / 2))) ") * ((cos . (x - (h / 2))) "))) by NEWTON:6
.= (((sin . (x + (h / 2))) |^ (2 + 1)) * (((cos . (x + (h / 2))) ") * ((cos . (x + (h / 2))) "))) - ((((sin . (x - (h / 2))) * (sin . (x - (h / 2)))) * (sin . (x - (h / 2)))) * (((cos . (x - (h / 2))) ") * ((cos . (x - (h / 2))) "))) by NEWTON:6
.= (((sin . (x + (h / 2))) |^ 3) * ((((cos . (x + (h / 2))) ") |^ 1) * ((cos . (x + (h / 2))) "))) - ((((sin . (x - (h / 2))) * (sin . (x - (h / 2)))) * (sin . (x - (h / 2)))) * (((cos . (x - (h / 2))) ") * ((cos . (x - (h / 2))) ")))
.= (((sin . (x + (h / 2))) |^ 3) * (((cos . (x + (h / 2))) ") |^ (1 + 1))) - ((((sin . (x - (h / 2))) * (sin . (x - (h / 2)))) * (sin . (x - (h / 2)))) * (((cos . (x - (h / 2))) ") * ((cos . (x - (h / 2))) "))) by NEWTON:6
.= (((sin . (x + (h / 2))) |^ 3) * (((cos . (x + (h / 2))) ") |^ 2)) - (((((sin . (x - (h / 2))) |^ 1) * (sin . (x - (h / 2)))) * (sin . (x - (h / 2)))) * (((cos . (x - (h / 2))) ") * ((cos . (x - (h / 2))) ")))
.= (((sin . (x + (h / 2))) |^ 3) * (((cos . (x + (h / 2))) ") |^ 2)) - ((((sin . (x - (h / 2))) |^ (1 + 1)) * (sin . (x - (h / 2)))) * (((cos . (x - (h / 2))) ") * ((cos . (x - (h / 2))) "))) by NEWTON:6
.= (((sin . (x + (h / 2))) |^ 3) * (((cos . (x + (h / 2))) ") |^ 2)) - (((sin . (x - (h / 2))) |^ (2 + 1)) * (((cos . (x - (h / 2))) ") * ((cos . (x - (h / 2))) "))) by NEWTON:6
.= (((sin . (x + (h / 2))) |^ 3) * (((cos . (x + (h / 2))) ") |^ 2)) - (((sin . (x - (h / 2))) |^ 3) * ((((cos . (x - (h / 2))) ") |^ 1) * ((cos . (x - (h / 2))) ")))
.= (((sin . (x + (h / 2))) |^ 3) * (((cos . (x + (h / 2))) ") |^ 2)) - (((sin . (x - (h / 2))) |^ 3) * (((cos . (x - (h / 2))) ") |^ (1 + 1))) by NEWTON:6
.= (((sin . (x + (h / 2))) |^ 3) * (((cos . (x + (h / 2))) ") |^ 2)) - (((sin . (x - (h / 2))) |^ 3) * (((cos . (x - (h / 2))) ") |^ 2)) ;
hence (cD (((tan (#) tan) (#) sin),h)) . x = (((sin . (x + (h / 2))) |^ 3) * (((cos . (x + (h / 2))) ") |^ 2)) - (((sin . (x - (h / 2))) |^ 3) * (((cos . (x - (h / 2))) ") |^ 2)) ; :: thesis: verum