let k, x0, x1, x2, x3, x4 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_different 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_different 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 & x1 <> 0 & x2 <> 0 & x3 <> 0 & x4 <> 0 ) ; :: thesis: ( not x0,x1,x2,x3,x4 are_mutually_different or [!f,x0,x1,x2,x3,x4!] = k / ((((x0 * x1) * x2) * x3) * x4) )
assume A3: x0,x1,x2,x3,x4 are_mutually_different ; :: thesis: [!f,x0,x1,x2,x3,x4!] = k / ((((x0 * x1) * x2) * x3) * x4)
then ( x0 <> x1 & x0 <> x2 & x0 <> x3 & x0 <> x4 & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 ) by ZFMISC_1:def 7;
then A4: ( x0,x1,x2,x3 are_mutually_different & x1,x2,x3,x4 are_mutually_different ) by ZFMISC_1:def 6;
A5: ( x0 - x1 <> 0 & x0 - x2 <> 0 & x0 - x3 <> 0 & x0 - x4 <> 0 & x1 - x2 <> 0 & x1 - x3 <> 0 & x1 - x4 <> 0 & x2 - x3 <> 0 & x2 - x4 <> 0 & x3 - x4 <> 0 ) by A3, ZFMISC_1:def 7;
[!f,x0,x1,x2,x3,x4!] = ((- (k / (((x0 * x1) * x2) * x3))) - [!f,x1,x2,x3,x4!]) / (x0 - x4) by A1, A2, A4, Th36
.= ((- (k / (((x0 * x1) * x2) * x3))) - (- (k / (((x1 * x2) * x3) * x4)))) / (x0 - x4) by A1, A2, A4, 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 A2, XCMPLX_1:92
.= ((- ((k * x4) / ((((x0 * x1) * x2) * x3) * x4))) + ((k * x0) / (x0 * (((x1 * x2) * x3) * x4)))) / (x0 - x4) by A2, XCMPLX_1:92
.= (((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_1:121
.= (k * (x0 - x4)) / (((((x0 * x1) * x2) * x3) * x4) * (x0 - x4)) by XCMPLX_1:79 ;
hence [!f,x0,x1,x2,x3,x4!] = k / ((((x0 * x1) * x2) * x3) * x4) by A5, XCMPLX_1:92; :: thesis: verum