let x0, x1 be Real; :: thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = 1 / ((sin x) ^2) ) & x0 <> x1 & sin x0 <> 0 & sin x1 <> 0 holds
[!f,x0,x1!] = ((((16 * (cos ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 - x0) / 2))) * (sin ((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 / ((sin x) ^2) ) & x0 <> x1 & sin x0 <> 0 & sin x1 <> 0 implies [!f,x0,x1!] = ((((16 * (cos ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 - x0) / 2))) * (sin ((x1 + x0) / 2))) / ((((cos (x0 + x1)) - (cos (x0 - x1))) ^2) * (x0 - x1)) )
assume that
A1: for x being Real holds f . x = 1 / ((sin x) ^2) and
x0 <> x1 and
A2: ( sin x0 <> 0 & sin x1 <> 0 ) ; :: thesis: [!f,x0,x1!] = ((((16 * (cos ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 - x0) / 2))) * (sin ((x1 + x0) / 2))) / ((((cos (x0 + x1)) - (cos (x0 - x1))) ^2) * (x0 - x1))
( f . x0 = 1 / ((sin x0) ^2) & f . x1 = 1 / ((sin x1) ^2) ) by A1;
then [!f,x0,x1!] = (((1 * ((sin x1) ^2)) - (1 * ((sin x0) ^2))) / (((sin x0) ^2) * ((sin x1) ^2))) / (x0 - x1) by A2, XCMPLX_1:130
.= ((((sin x1) ^2) - ((sin x0) ^2)) / (((sin x0) * (sin x1)) ^2)) / (x0 - x1)
.= ((((sin x1) ^2) - ((sin x0) ^2)) / ((- ((1 / 2) * ((cos (x0 + x1)) - (cos (x0 - x1))))) ^2)) / (x0 - x1) by SIN_COS4:29
.= ((((sin x1) ^2) - ((sin x0) ^2)) / ((1 / 4) * (((cos (x0 + x1)) - (cos (x0 - x1))) ^2))) / (x0 - x1)
.= (((((sin x1) ^2) - ((sin x0) ^2)) / (1 / 4)) / (((cos (x0 + x1)) - (cos (x0 - x1))) ^2)) / (x0 - x1) by XCMPLX_1:78
.= ((4 * (((sin x1) - (sin x0)) * ((sin x1) + (sin x0)))) / (((cos (x0 + x1)) - (cos (x0 - x1))) ^2)) / (x0 - x1)
.= ((4 * ((2 * ((cos ((x1 + x0) / 2)) * (sin ((x1 - x0) / 2)))) * ((sin x1) + (sin x0)))) / (((cos (x0 + x1)) - (cos (x0 - x1))) ^2)) / (x0 - x1) by SIN_COS4:16
.= ((4 * ((2 * ((cos ((x1 + x0) / 2)) * (sin ((x1 - x0) / 2)))) * (2 * ((cos ((x1 - x0) / 2)) * (sin ((x1 + x0) / 2)))))) / (((cos (x0 + x1)) - (cos (x0 - x1))) ^2)) / (x0 - x1) by SIN_COS4:15
.= ((((16 * (cos ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 - x0) / 2))) * (sin ((x1 + x0) / 2))) / ((((cos (x0 + x1)) - (cos (x0 - x1))) ^2) * (x0 - x1)) by XCMPLX_1:78 ;
hence [!f,x0,x1!] = ((((16 * (cos ((x1 + x0) / 2))) * (sin ((x1 - x0) / 2))) * (cos ((x1 - x0) / 2))) * (sin ((x1 + x0) / 2))) / ((((cos (x0 + x1)) - (cos (x0 - x1))) ^2) * (x0 - x1)) ; :: thesis: verum