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_1:130
.= ((((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_COS4:32
.= ((((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_1:78
.= ((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_COS4:18
.= ((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_COS4:17
.= ((((((- 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