let x0, x1, x2, x3, k be Real; :: thesis: 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; :: thesis: ( ( 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 ; :: thesis: ( 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 ; :: thesis: [!f,x0,x1,x2,x3!] = - (k / (((x0 * x1) * x2) * x3))
then A6: x1 <> x2 by ZFMISC_1:def 6;
( x1 <> x3 & x2 <> x3 ) by ;
then A7: x1,x2,x3 are_mutually_distinct by ;
A8: x0 - x3 <> 0 by ;
( x0 <> x1 & x0 <> x2 ) by ;
then x0,x1,x2 are_mutually_distinct by ;
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
.= (((k * x3) / (((x0 * x1) * x2) * x3)) - ((k * x0) / (x0 * ((x1 * x2) * x3)))) / (x0 - x3) by
.= (((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 ;
hence [!f,x0,x1,x2,x3!] = - (k / (((x0 * x1) * x2) * x3)) by XCMPLX_1:187; :: thesis: verum