let x0, x1, k be Real; for f being Function of REAL,REAL st ( for x being Real holds f . x = k / x ) & x0 <> x1 & x0 <> 0 & x1 <> 0 holds
[!f,x0,x1!] = - (k / (x0 * x1))
let f be Function of REAL,REAL; ( ( for x being Real holds f . x = k / x ) & x0 <> x1 & x0 <> 0 & x1 <> 0 implies [!f,x0,x1!] = - (k / (x0 * x1)) )
assume that
A1:
for x being Real holds f . x = k / x
and
A2:
x0 <> x1
and
A3:
x0 <> 0
and
A4:
x1 <> 0
; [!f,x0,x1!] = - (k / (x0 * x1))
A5:
x1 - x0 <> 0
by A2;
[!f,x0,x1!] =
((k / x0) - (f . x1)) / (x0 - x1)
by A1
.=
((k / x0) - (k / x1)) / (x0 - x1)
by A1
.=
(((k * x1) / (x0 * x1)) - (k / x1)) / (x0 - x1)
by A4, XCMPLX_1:91
.=
(((k * x1) / (x0 * x1)) - ((k * x0) / (x0 * x1))) / (x0 - x1)
by A3, XCMPLX_1:91
.=
(((k * x1) - (k * x0)) / (x0 * x1)) / (x0 - x1)
by XCMPLX_1:120
.=
((k * (x1 - x0)) / (x0 * x1)) / (- (x1 - x0))
.=
- (((k * (x1 - x0)) / (x0 * x1)) / (x1 - x0))
by XCMPLX_1:188
.=
- (((k * (x1 - x0)) / (x1 - x0)) / (x0 * x1))
by XCMPLX_1:48
;
hence
[!f,x0,x1!] = - (k / (x0 * x1))
by A5, XCMPLX_1:89; verum