let x0, x1, x2, x3, x4, 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 & x4 <> 0 & x0,x1,x2,x3,x4 are_mutually_distinct holds
[!f,x0,x1,x2,x3,x4!] = k / ((((x0 * x1) * x2) * x3) * x4)
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 & x4 <> 0 & x0,x1,x2,x3,x4 are_mutually_distinct implies [!f,x0,x1,x2,x3,x4!] = k / ((((x0 * x1) * x2) * x3) * x4) )
assume that
A1: for x being Real holds f . x = k / x and
A2: x0 <> 0 and
A3: ( x1 <> 0 & x2 <> 0 & x3 <> 0 ) and
A4: x4 <> 0 ; :: thesis: ( not x0,x1,x2,x3,x4 are_mutually_distinct or [!f,x0,x1,x2,x3,x4!] = k / ((((x0 * x1) * x2) * x3) * x4) )
assume A5: x0,x1,x2,x3,x4 are_mutually_distinct ; :: thesis: [!f,x0,x1,x2,x3,x4!] = k / ((((x0 * x1) * x2) * x3) * x4)
then A6: ( x1 <> x2 & x1 <> x3 ) by ZFMISC_1:def 7;
A7: x3 <> x4 by A5, ZFMISC_1:def 7;
A8: x2 <> x3 by A5, ZFMISC_1:def 7;
A9: x0 - x4 <> 0 by A5, ZFMISC_1:def 7;
A10: x0 <> x3 by A5, ZFMISC_1:def 7;
( x1 <> x4 & x2 <> x4 ) by A5, ZFMISC_1:def 7;
then A11: x1,x2,x3,x4 are_mutually_distinct by A6, A8, A7, ZFMISC_1:def 6;
( x0 <> x1 & x0 <> x2 ) by A5, ZFMISC_1:def 7;
then x0,x1,x2,x3 are_mutually_distinct by A10, A6, A8, ZFMISC_1:def 6;
then [!f,x0,x1,x2,x3,x4!] = ((- (k / (((x0 * x1) * x2) * x3))) - [!f,x1,x2,x3,x4!]) / (x0 - x4) by A1, A2, A3, Th36
.= ((- (k / (((x0 * x1) * x2) * x3))) - (- (k / (((x1 * x2) * x3) * x4)))) / (x0 - x4) by A1, A3, A4, A11, Th36
.= ((- (k / (((x0 * x1) * x2) * x3))) + (k / (((x1 * x2) * x3) * x4))) / (x0 - x4)
.= ((- ((k * x4) / ((((x0 * x1) * x2) * x3) * x4))) + (k / (((x1 * x2) * x3) * x4))) / (x0 - x4) by A4, XCMPLX_1:91
.= ((- ((k * x4) / ((((x0 * x1) * x2) * x3) * x4))) + ((k * x0) / (x0 * (((x1 * x2) * x3) * x4)))) / (x0 - x4) by A2, XCMPLX_1:91
.= (((k * x0) / ((((x0 * x1) * x2) * x3) * x4)) - ((k * x4) / ((((x0 * x1) * x2) * x3) * x4))) / (x0 - x4)
.= (((k * x0) - (k * x4)) / ((((x0 * x1) * x2) * x3) * x4)) / (x0 - x4) by XCMPLX_1:120
.= (k * (x0 - x4)) / (((((x0 * x1) * x2) * x3) * x4) * (x0 - x4)) by XCMPLX_1:78 ;
hence [!f,x0,x1,x2,x3,x4!] = k / ((((x0 * x1) * x2) * x3) * x4) by A9, XCMPLX_1:91; :: thesis: verum