let x0, x1 be Real; :: thesis: for f being Function of REAL ,REAL st ( for x being Real holds f . x = 1 / ((cos x) ^2 ) ) & x0 <> x1 & cos x0 <> 0 & cos x1 <> 0 holds
[!f,x0,x1!] = ((((((- 16) * (sin ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 + x0) / 2))) * (cos ((x1 - x0) / 2))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2 )) / (x0 - x1)
let f be Function of REAL ,REAL ; :: thesis: ( ( for x being Real holds f . x = 1 / ((cos x) ^2 ) ) & x0 <> x1 & cos x0 <> 0 & cos x1 <> 0 implies [!f,x0,x1!] = ((((((- 16) * (sin ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 + x0) / 2))) * (cos ((x1 - x0) / 2))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2 )) / (x0 - x1) )
assume that
A1: for x being Real holds f . x = 1 / ((cos x) ^2 ) and
x0 <> x1 and
A2: ( cos x0 <> 0 & cos x1 <> 0 ) ; :: thesis: [!f,x0,x1!] = ((((((- 16) * (sin ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 + x0) / 2))) * (cos ((x1 - x0) / 2))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2 )) / (x0 - x1)
( f . x0 = 1 / ((cos x0) ^2 ) & f . x1 = 1 / ((cos x1) ^2 ) ) by A1;
then [!f,x0,x1!] = (((1 * ((cos x1) ^2 )) - (1 * ((cos x0) ^2 ))) / (((cos x0) ^2 ) * ((cos x1) ^2 ))) / (x0 - x1) by A2, XCMPLX_1:131
.= ((((cos x1) ^2 ) - ((cos x0) ^2 )) / (((cos x0) * (cos x1)) ^2 )) / (x0 - x1)
.= ((((cos x1) ^2 ) - ((cos x0) ^2 )) / (((1 / 2) * ((cos (x0 + x1)) + (cos (x0 - x1)))) ^2 )) / (x0 - x1) by SIN_COS4:36
.= ((((cos x1) ^2 ) - ((cos x0) ^2 )) / ((1 / 4) * (((cos (x0 + x1)) + (cos (x0 - x1))) ^2 ))) / (x0 - x1)
.= (((((cos x1) ^2 ) - ((cos x0) ^2 )) / (1 / 4)) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2 )) / (x0 - x1) by XCMPLX_1:79
.= ((4 * (((cos x1) - (cos x0)) * ((cos x1) + (cos x0)))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2 )) / (x0 - x1)
.= ((4 * ((- (2 * ((sin ((x1 + x0) / 2)) * (sin ((x1 - x0) / 2))))) * ((cos x1) + (cos x0)))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2 )) / (x0 - x1) by SIN_COS4:22
.= ((4 * ((- (2 * ((sin ((x1 + x0) / 2)) * (sin ((x1 - x0) / 2))))) * (2 * ((cos ((x1 + x0) / 2)) * (cos ((x1 - x0) / 2)))))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2 )) / (x0 - x1) by SIN_COS4:21
.= ((((((- 16) * (sin ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 + x0) / 2))) * (cos ((x1 - x0) / 2))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2 )) / (x0 - x1) ;
hence [!f,x0,x1!] = ((((((- 16) * (sin ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 + x0) / 2))) * (cos ((x1 - x0) / 2))) / (((cos (x0 + x1)) + (cos (x0 - x1))) ^2 )) / (x0 - x1) ; :: thesis: verum