let x0, x1 be Real; :: thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) cos) . x ) & x0 in dom tan & x1 in dom tan holds
[!f,x0,x1!] = ((sin x0) - (sin x1)) / (x0 - x1)
let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = (tan (#) cos) . x ) & x0 in dom tan & x1 in dom tan implies [!f,x0,x1!] = ((sin x0) - (sin x1)) / (x0 - x1) )
assume that
A1: for x being Real holds f . x = (tan (#) cos) . x and
A2: ( x0 in dom tan & x1 in dom tan ) ; :: thesis: [!f,x0,x1!] = ((sin x0) - (sin x1)) / (x0 - x1)
A3: f . x0 = (tan (#) cos) . x0 by A1;
f . x1 = (tan (#) cos) . x1 by A1;
then [!f,x0,x1!] = (((tan . x0) * (cos . x0)) - ((tan (#) cos) . x1)) / (x0 - x1) by A3, VALUED_1:5
.= (((tan . x0) * (cos . x0)) - ((tan . x1) * (cos . x1))) / (x0 - x1) by VALUED_1:5
.= ((((sin . x0) * ((cos . x0) ")) * (cos . x0)) - ((tan . x1) * (cos . x1))) / (x0 - x1) by A2, RFUNCT_1:def 1
.= ((((sin x0) / (cos x0)) * (cos x0)) - (((sin x1) / (cos x1)) * (cos x1))) / (x0 - x1) by A2, RFUNCT_1:def 1
.= (((sin x0) / ((cos x0) / (cos x0))) - (((sin x1) / (cos x1)) * (cos x1))) / (x0 - x1) by XCMPLX_1:82
.= (((sin x0) / ((cos x0) * (1 / (cos x0)))) - ((sin x1) / ((cos x1) / (cos x1)))) / (x0 - x1) by XCMPLX_1:82
.= (((sin x0) / 1) - ((sin x1) / ((cos x1) * (1 / (cos x1))))) / (x0 - x1) by A2, FDIFF_8:1, XCMPLX_1:106
.= (((sin x0) / 1) - ((sin x1) / 1)) / (x0 - x1) by A2, FDIFF_8:1, XCMPLX_1:106
.= ((sin x0) - (sin x1)) / (x0 - x1) ;
hence [!f,x0,x1!] = ((sin x0) - (sin x1)) / (x0 - x1) ; :: thesis: verum