let x0, x1 be Real; :: thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (cot (#) cos) . x ) & x0 in dom cot & x1 in dom cot holds
[!f,x0,x1!] = ((((1 / (sin x0)) - (sin x0)) - (1 / (sin x1))) + (sin x1)) / (x0 - x1)
let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = (cot (#) cos) . x ) & x0 in dom cot & x1 in dom cot implies [!f,x0,x1!] = ((((1 / (sin x0)) - (sin x0)) - (1 / (sin x1))) + (sin x1)) / (x0 - x1) )
assume that
A1: for x being Real holds f . x = (cot (#) cos) . x and
A2: ( x0 in dom cot & x1 in dom cot ) ; :: thesis: [!f,x0,x1!] = ((((1 / (sin x0)) - (sin x0)) - (1 / (sin x1))) + (sin x1)) / (x0 - x1)
A3: f . x0 = (cot (#) cos) . x0 by A1;
f . x1 = (cot (#) cos) . x1 by A1;
then [!f,x0,x1!] = (((cot . x0) * (cos . x0)) - ((cot (#) cos) . x1)) / (x0 - x1) by A3, VALUED_1:5
.= (((cot . x0) * (cos . x0)) - ((cot . x1) * (cos . x1))) / (x0 - x1) by VALUED_1:5
.= ((((cos . x0) * ((sin . x0) ")) * (cos . x0)) - ((cot . x1) * (cos . x1))) / (x0 - x1) by A2, RFUNCT_1:def 1
.= ((((cos x0) / (sin x0)) * (cos x0)) - (((cos x1) / (sin x1)) * (cos x1))) / (x0 - x1) by A2, RFUNCT_1:def 1
.= (((cos x0) / ((sin x0) / (cos x0))) - (((cos x1) / (sin x1)) * (cos x1))) / (x0 - x1) by XCMPLX_1:82
.= (((cos x0) / ((sin x0) / (cos x0))) - ((cos x1) / ((sin x1) / (cos x1)))) / (x0 - x1) by XCMPLX_1:82
.= ((((cos x0) * (cos x0)) / (sin x0)) - ((cos x1) / ((sin x1) / (cos x1)))) / (x0 - x1) by XCMPLX_1:77
.= ((((cos x0) * (cos x0)) / (sin x0)) - (((cos x1) * (cos x1)) / (sin x1))) / (x0 - x1) by XCMPLX_1:77
.= (((1 - ((sin x0) * (sin x0))) / (sin x0)) - (((cos x1) * (cos x1)) / (sin x1))) / (x0 - x1) by SIN_COS4:5
.= (((1 / (sin x0)) - (((sin x0) * (sin x0)) / (sin x0))) - ((1 - ((sin x1) * (sin x1))) / (sin x1))) / (x0 - x1) by SIN_COS4:5
.= (((1 / (sin x0)) - (sin x0)) - ((1 / (sin x1)) - (((sin x1) * (sin x1)) / (sin x1)))) / (x0 - x1) by A2, FDIFF_8:2, XCMPLX_1:89
.= (((1 / (sin x0)) - (sin x0)) - ((1 / (sin x1)) - (sin x1))) / (x0 - x1) by A2, FDIFF_8:2, XCMPLX_1:89
.= ((((1 / (sin x0)) - (sin x0)) - (1 / (sin x1))) + (sin x1)) / (x0 - x1) ;
hence [!f,x0,x1!] = ((((1 / (sin x0)) - (sin x0)) - (1 / (sin x1))) + (sin x1)) / (x0 - x1) ; :: thesis: verum