let x0, x1 be Real; :: thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = (tan (#) tan) . x ) & x0 in dom tan & x1 in dom tan holds
[!f,x0,x1!] = (((cos x1) ^2) - ((cos x0) ^2)) / ((((cos x0) * (cos x1)) ^2) * (x0 - x1))
let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = (tan (#) tan) . x ) & x0 in dom tan & x1 in dom tan implies [!f,x0,x1!] = (((cos x1) ^2) - ((cos x0) ^2)) / ((((cos x0) * (cos x1)) ^2) * (x0 - x1)) )
assume that
A1: for x being Real holds f . x = (tan (#) tan) . x and
A2: ( x0 in dom tan & x1 in dom tan ) ; :: thesis: [!f,x0,x1!] = (((cos x1) ^2) - ((cos x0) ^2)) / ((((cos x0) * (cos x1)) ^2) * (x0 - x1))
A3: ( cos x0 <> 0 & cos x1 <> 0 ) by A2, FDIFF_8:1;
A4: f . x0 = (tan (#) tan) . x0 by A1;
f . x1 = (tan (#) tan) . x1 by A1;
then [!f,x0,x1!] = (((tan . x0) * (tan . x0)) - ((tan (#) tan) . x1)) / (x0 - x1) by A4, VALUED_1:5
.= (((tan . x0) * (tan . x0)) - ((tan . x1) * (tan . x1))) / (x0 - x1) by VALUED_1:5
.= ((((sin . x0) * ((cos . x0) ")) * (tan . x0)) - ((tan . x1) * (tan . x1))) / (x0 - x1) by A2, RFUNCT_1:def 1
.= ((((sin . x0) * ((cos . x0) ")) * ((sin . x0) * ((cos . x0) "))) - ((tan . x1) * (tan . x1))) / (x0 - x1) by A2, RFUNCT_1:def 1
.= ((((sin . x0) * ((cos . x0) ")) * ((sin . x0) * ((cos . x0) "))) - (((sin . x1) * ((cos . x1) ")) * (tan . x1))) / (x0 - x1) by A2, RFUNCT_1:def 1
.= (((tan x0) ^2) - ((tan x1) ^2)) / (x0 - x1) by A2, RFUNCT_1:def 1
.= (((tan x0) - (tan x1)) * ((tan x0) + (tan x1))) / (x0 - x1)
.= (((sin (x0 - x1)) / ((cos x0) * (cos x1))) * ((tan x0) + (tan x1))) / (x0 - x1) by A3, SIN_COS4:20
.= (((sin (x0 - x1)) / ((cos x0) * (cos x1))) * ((sin (x0 + x1)) / ((cos x0) * (cos x1)))) / (x0 - x1) by A3, SIN_COS4:19
.= (((sin (x0 + x1)) * (sin (x0 - x1))) / (((cos x0) * (cos x1)) ^2)) / (x0 - x1) by XCMPLX_1:76
.= ((((cos x1) ^2) - ((cos x0) ^2)) / (((cos x0) * (cos x1)) ^2)) / (x0 - x1) by SIN_COS4:38
.= (((cos x1) ^2) - ((cos x0) ^2)) / ((((cos x0) * (cos x1)) ^2) * (x0 - x1)) by XCMPLX_1:78 ;
hence [!f,x0,x1!] = (((cos x1) ^2) - ((cos x0) ^2)) / ((((cos x0) * (cos x1)) ^2) * (x0 - x1)) ; :: thesis: verum