let k, x0, x1 be Real; :: thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = k / (x ^2) ) & x0 <> x1 & x0 <> 0 & x1 <> 0 holds
[!f,x0,x1!] = - ((k / (x0 * x1)) * ((1 / x0) + (1 / x1)))
let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = k / (x ^2) ) & x0 <> x1 & x0 <> 0 & x1 <> 0 implies [!f,x0,x1!] = - ((k / (x0 * x1)) * ((1 / x0) + (1 / x1))) )
assume that
A1: for x being Real holds f . x = k / (x ^2) and
A2: ( x0 <> x1 & x0 <> 0 & x1 <> 0 ) ; :: thesis: [!f,x0,x1!] = - ((k / (x0 * x1)) * ((1 / x0) + (1 / x1)))
A3: x1 - x0 <> 0 by A2;
( f . x0 = k / (x0 ^2) & f . x1 = k / (x1 ^2) ) by A1;
then [!f,x0,x1!] = (k * ((1 / (x0 ^2)) - (1 / (x1 ^2)))) / (x0 - x1)
.= (k * (((1 * (x1 ^2)) - (1 * (x0 ^2))) / ((x0 ^2) * (x1 ^2)))) / (x0 - x1) by A2, XCMPLX_1:130
.= k * ((((x1 - x0) * (x1 + x0)) / ((x0 ^2) * (x1 ^2))) / (- (x1 - x0)))
.= k * (- ((((x1 - x0) * (x1 + x0)) / ((x0 ^2) * (x1 ^2))) / (x1 - x0))) by XCMPLX_1:188
.= - (k * ((((x1 - x0) * (x1 + x0)) / (x1 - x0)) / ((x0 ^2) * (x1 ^2))))
.= - (k * ((x1 + x0) / (((x0 * x0) * x1) * x1))) by A3, XCMPLX_1:89
.= - (k * ((x1 / (x1 * ((x0 * x0) * x1))) + (x0 / (x0 * ((x0 * x1) * x1)))))
.= - (k * (((1 / ((x0 * x0) * x1)) * (x1 / x1)) + (x0 / (x0 * ((x0 * x1) * x1))))) by XCMPLX_1:103
.= - (k * (((1 / ((x0 * x0) * x1)) * (x1 / x1)) + ((1 / ((x0 * x1) * x1)) * (x0 / x0)))) by XCMPLX_1:103
.= - (k * (((1 / ((x0 * x0) * x1)) * 1) + ((1 / ((x0 * x1) * x1)) * (x0 / x0)))) by A2, XCMPLX_1:60
.= - (k * ((1 / (x0 * (x0 * x1))) + (1 / ((x0 * x1) * x1)))) by A2, XCMPLX_1:60
.= - (k * (((1 / x0) * (1 / (x0 * x1))) + (1 / ((x0 * x1) * x1)))) by XCMPLX_1:102
.= - (k * (((1 / x0) * (1 / (x0 * x1))) + ((1 / (x0 * x1)) * (1 / x1)))) by XCMPLX_1:102
.= - ((k / (x0 * x1)) * ((1 / x0) + (1 / x1))) ;
hence [!f,x0,x1!] = - ((k / (x0 * x1)) * ((1 / x0) + (1 / x1))) ; :: thesis: verum