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