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;
A5: f . x1 = (tan (#) tan ) . x1 by A1;
[!f,x0,x1!] = (((tan . x0) * (tan . x0)) - ((tan (#) tan ) . x1)) / (x0 - x1) by A4, A5, 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 4
.= ((((sin . x0) * ((cos . x0) " )) * ((sin . x0) * ((cos . x0) " ))) - ((tan . x1) * (tan . x1))) / (x0 - x1) by A2, RFUNCT_1:def 4
.= ((((sin . x0) * ((cos . x0) " )) * ((sin . x0) * ((cos . x0) " ))) - (((sin . x1) * ((cos . x1) " )) * (tan . x1))) / (x0 - x1) by A2, RFUNCT_1:def 4
.= (((tan x0) ^2 ) - ((tan x1) ^2 )) / (x0 - x1) by A2, RFUNCT_1:def 4
.= (((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_COS4:24
.= (((sin (x0 - x1)) / ((cos x0) * (cos x1))) * ((sin (x0 + x1)) / ((cos x0) * (cos x1)))) / (x0 - x1) by A3, SIN_COS4:23
.= (((sin (x0 + x1)) * (sin (x0 - x1))) / (((cos x0) * (cos x1)) ^2 )) / (x0 - x1) by XCMPLX_1:77
.= ((((cos x1) ^2 ) - ((cos x0) ^2 )) / (((cos x0) * (cos x1)) ^2 )) / (x0 - x1) by SIN_COS4:42
.= (((cos x1) ^2 ) - ((cos x0) ^2 )) / ((((cos x0) * (cos x1)) ^2 ) * (x0 - x1)) by XCMPLX_1:79 ;
hence [!f,x0,x1!] = (((cos x1) ^2 ) - ((cos x0) ^2 )) / ((((cos x0) * (cos x1)) ^2 ) * (x0 - x1)) ; :: thesis: verum