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

proof

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

proof

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

end;

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

end;

then (bD (((cot (#) cot) (#) cos),h)) . x = (((cot (#) cot) (#) cos) . x) - (((cot (#) cot) (#) cos) . (x - h)) by DIFF_1:38
.= (((cot (#) cot) . x) * (cos . x)) - (((cot (#) cot) (#) cos) . (x - h)) by VALUED_1:5
.= (((cot (#) cot) . x) * (cos . x)) - (((cot (#) cot) . (x - h)) * (cos . (x - h))) by VALUED_1:5
.= (((cot . x) * (cot . x)) * (cos . x)) - (((cot (#) cot) . (x - h)) * (cos . (x - h))) by VALUED_1:5
.= (((cot . x) * (cot . x)) * (cos . x)) - (((cot . (x - h)) * (cot . (x - h))) * (cos . (x - h))) by VALUED_1:5
.= ((((cos . x) * ((sin . x) ")) * (cot . x)) * (cos . x)) - (((cot . (x - h)) * (cot . (x - h))) * (cos . (x - h))) by A1, RFUNCT_1:def 1
.= ((((cos . x) * ((sin . x) ")) * ((cos . x) * ((sin . x) "))) * (cos . x)) - (((cot . (x - h)) * (cot . (x - h))) * (cos . (x - h))) by A1, RFUNCT_1:def 1
.= ((((cos . x) * ((sin . x) ")) * ((cos . x) * ((sin . x) "))) * (cos . x)) - ((((cos . (x - h)) * ((sin . (x - h)) ")) * (cot . (x - h))) * (cos . (x - h))) by A1, RFUNCT_1:def 1
.= ((((cos . x) * ((sin . x) ")) * ((cos . x) * ((sin . x) "))) * (cos . x)) - ((((cos . (x - h)) * ((sin . (x - h)) ")) * ((cos . (x - h)) * ((sin . (x - h)) "))) * (cos . (x - h))) by A1, RFUNCT_1:def 1
.= ((((cos . x) * (cos . x)) * (cos . x)) * (((sin . x) ") * ((sin . x) "))) - ((((cos . (x - h)) * (cos . (x - h))) * (cos . (x - h))) * (((sin . (x - h)) ") * ((sin . (x - h)) ")))
.= (((((cos . x) |^ 1) * (cos . x)) * (cos . x)) * (((sin . x) ") * ((sin . x) "))) - ((((cos . (x - h)) * (cos . (x - h))) * (cos . (x - h))) * (((sin . (x - h)) ") * ((sin . (x - h)) ")))
.= ((((cos . x) |^ (1 + 1)) * (cos . x)) * (((sin . x) ") * ((sin . x) "))) - ((((cos . (x - h)) * (cos . (x - h))) * (cos . (x - h))) * (((sin . (x - h)) ") * ((sin . (x - h)) "))) by NEWTON:6
.= (((cos . x) |^ (2 + 1)) * (((sin . x) ") * ((sin . x) "))) - ((((cos . (x - h)) * (cos . (x - h))) * (cos . (x - h))) * (((sin . (x - h)) ") * ((sin . (x - h)) "))) by NEWTON:6
.= (((cos . x) |^ 3) * ((((sin . x) ") |^ 1) * ((sin . x) "))) - ((((cos . (x - h)) * (cos . (x - h))) * (cos . (x - h))) * (((sin . (x - h)) ") * ((sin . (x - h)) ")))
.= (((cos . x) |^ 3) * (((sin . x) ") |^ (1 + 1))) - ((((cos . (x - h)) * (cos . (x - h))) * (cos . (x - h))) * (((sin . (x - h)) ") * ((sin . (x - h)) "))) by NEWTON:6
.= (((cos . x) |^ 3) * (((sin . x) ") |^ 2)) - (((((cos . (x - h)) |^ 1) * (cos . (x - h))) * (cos . (x - h))) * (((sin . (x - h)) ") * ((sin . (x - h)) ")))
.= (((cos . x) |^ 3) * (((sin . x) ") |^ 2)) - ((((cos . (x - h)) |^ (1 + 1)) * (cos . (x - h))) * (((sin . (x - h)) ") * ((sin . (x - h)) "))) by NEWTON:6
.= (((cos . x) |^ 3) * (((sin . x) ") |^ 2)) - (((cos . (x - h)) |^ (2 + 1)) * (((sin . (x - h)) ") * ((sin . (x - h)) "))) by NEWTON:6
.= (((cos . x) |^ 3) * (((sin . x) ") |^ 2)) - (((cos . (x - h)) |^ 3) * ((((sin . (x - h)) ") |^ 1) * ((sin . (x - h)) ")))
.= (((cos . x) |^ 3) * (((sin . x) ") |^ 2)) - (((cos . (x - h)) |^ 3) * (((sin . (x - h)) ") |^ (1 + 1))) by NEWTON:6
.= (((cos . x) |^ 3) * (((sin . x) ") |^ 2)) - (((cos . (x - h)) |^ 3) * (((sin . (x - h)) ") |^ 2)) ;
hence (bD (((cot (#) cot) (#) cos),h)) . x = (((cos . x) |^ 3) * (((sin . x) ") |^ 2)) - (((cos . (x - h)) |^ 3) * (((sin . (x - h)) ") |^ 2)) ; :: thesis: verum