let x0, x1, x2, x3, k be Real; for f being Function of REAL,REAL st ( for x being Real holds f . x = k / x ) & x0 <> 0 & x1 <> 0 & x2 <> 0 & x3 <> 0 & x0,x1,x2,x3 are_mutually_distinct holds
[!f,x0,x1,x2,x3!] = - (k / (((x0 * x1) * x2) * x3))
let f be Function of REAL,REAL; ( ( for x being Real holds f . x = k / x ) & x0 <> 0 & x1 <> 0 & x2 <> 0 & x3 <> 0 & x0,x1,x2,x3 are_mutually_distinct implies [!f,x0,x1,x2,x3!] = - (k / (((x0 * x1) * x2) * x3)) )
assume that
A1:
for x being Real holds f . x = k / x
and
A2:
x0 <> 0
and
A3:
( x1 <> 0 & x2 <> 0 )
and
A4:
x3 <> 0
; ( not x0,x1,x2,x3 are_mutually_distinct or [!f,x0,x1,x2,x3!] = - (k / (((x0 * x1) * x2) * x3)) )
assume A5:
x0,x1,x2,x3 are_mutually_distinct
; [!f,x0,x1,x2,x3!] = - (k / (((x0 * x1) * x2) * x3))
then A6:
x1 <> x2
by ZFMISC_1:def 6;
( x1 <> x3 & x2 <> x3 )
by A5, ZFMISC_1:def 6;
then A7:
x1,x2,x3 are_mutually_distinct
by A6, ZFMISC_1:def 5;
A8:
x0 - x3 <> 0
by A5, ZFMISC_1:def 6;
( x0 <> x1 & x0 <> x2 )
by A5, ZFMISC_1:def 6;
then
x0,x1,x2 are_mutually_distinct
by A6, ZFMISC_1:def 5;
then [!f,x0,x1,x2,x3!] =
((k / ((x0 * x1) * x2)) - [!f,x1,x2,x3!]) / (x0 - x3)
by A1, A2, A3, Th35
.=
((k / ((x0 * x1) * x2)) - (k / ((x1 * x2) * x3))) / (x0 - x3)
by A1, A3, A4, A7, Th35
.=
(((k * x3) / (((x0 * x1) * x2) * x3)) - (k / ((x1 * x2) * x3))) / (x0 - x3)
by A4, XCMPLX_1:91
.=
(((k * x3) / (((x0 * x1) * x2) * x3)) - ((k * x0) / (x0 * ((x1 * x2) * x3)))) / (x0 - x3)
by A2, XCMPLX_1:91
.=
(((k * x3) - (k * x0)) / (((x0 * x1) * x2) * x3)) / (x0 - x3)
by XCMPLX_1:120
.=
((- k) * (x0 - x3)) / ((((x0 * x1) * x2) * x3) * (x0 - x3))
by XCMPLX_1:78
.=
(- k) / (((x0 * x1) * x2) * x3)
by A8, XCMPLX_1:91
;
hence
[!f,x0,x1,x2,x3!] = - (k / (((x0 * x1) * x2) * x3))
by XCMPLX_1:187; verum