let k, x0, x1, x2, x3 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_different 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_different 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 & x1 <> 0 & x2 <> 0 & x3 <> 0 ) ; :: thesis: ( not x0,x1,x2,x3 are_mutually_different or [!f,x0,x1,x2,x3!] = - (k / (((x0 * x1) * x2) * x3)) )
assume A3: x0,x1,x2,x3 are_mutually_different ; :: thesis: [!f,x0,x1,x2,x3!] = - (k / (((x0 * x1) * x2) * x3))
then ( x0 <> x1 & x0 <> x2 & x0 <> x3 & x1 <> x2 & x1 <> x3 & x2 <> x3 ) by ZFMISC_1:def 6;
then A4: ( x0,x1,x2 are_mutually_different & x1,x2,x3 are_mutually_different ) by ZFMISC_1:def 5;
A5: ( x0 - x1 <> 0 & x0 - x2 <> 0 & x0 - x3 <> 0 & x1 - x2 <> 0 & x1 - x3 <> 0 & x2 - x3 <> 0 ) by A3, ZFMISC_1:def 6;
[!f,x0,x1,x2,x3!] = ((k / ((x0 * x1) * x2)) - [!f,x1,x2,x3!]) / (x0 - x3) by A1, A2, A4, Th35
.= ((k / ((x0 * x1) * x2)) - (k / ((x1 * x2) * x3))) / (x0 - x3) by A1, A2, A4, Th35
.= (((k * x3) / (((x0 * x1) * x2) * x3)) - (k / ((x1 * x2) * x3))) / (x0 - x3) by A2, XCMPLX_1:92
.= (((k * x3) / (((x0 * x1) * x2) * x3)) - ((k * x0) / (x0 * ((x1 * x2) * x3)))) / (x0 - x3) by A2, XCMPLX_1:92
.= (((k * x3) - (k * x0)) / (((x0 * x1) * x2) * x3)) / (x0 - x3) by XCMPLX_1:121
.= ((- k) * (x0 - x3)) / ((((x0 * x1) * x2) * x3) * (x0 - x3)) by XCMPLX_1:79
.= (- k) / (((x0 * x1) * x2) * x3) by A5, XCMPLX_1:92 ;
hence [!f,x0,x1,x2,x3!] = - (k / (((x0 * x1) * x2) * x3)) by XCMPLX_1:188; :: thesis: verum