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))
A3: ( f . x0 = 1 / ((sin x0) ^2 ) & f . x1 = 1 / ((sin x1) ^2 ) ) by A1;
[!f,x0,x1!] = (((1 * ((sin x1) ^2 )) - (1 * ((sin x0) ^2 ))) / (((sin x0) ^2 ) * ((sin x1) ^2 ))) / (x0 - x1) by A2, A3, XCMPLX_1:131
.= ((((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_COS4:33
.= ((((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_1:79
.= ((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_COS4:20
.= ((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_COS4:19
.= ((((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_1:79 ;
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