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 ^2) ) & x0 <> 0 & x1 <> 0 & x2 <> 0 & x3 <> 0 & x0,x1,x2,x3 are_mutually_distinct holds
[!f,x0,x1,x2,x3!] = (k * (((1 / ((x1 * x2) * x0)) * (((1 / x0) + (1 / x2)) + (1 / x1))) - ((1 / ((x2 * x1) * x3)) * (((1 / x3) + (1 / x1)) + (1 / x2))))) / (x0 - x3)
let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = k / (x ^2) ) & x0 <> 0 & x1 <> 0 & x2 <> 0 & x3 <> 0 & x0,x1,x2,x3 are_mutually_distinct implies [!f,x0,x1,x2,x3!] = (k * (((1 / ((x1 * x2) * x0)) * (((1 / x0) + (1 / x2)) + (1 / x1))) - ((1 / ((x2 * x1) * x3)) * (((1 / x3) + (1 / x1)) + (1 / x2))))) / (x0 - x3) )
assume that
A1: for x being Real holds f . x = k / (x ^2) and
A2: ( x0 <> 0 & x1 <> 0 & x2 <> 0 & x3 <> 0 ) and
A3: x0,x1,x2,x3 are_mutually_distinct ; :: thesis: [!f,x0,x1,x2,x3!] = (k * (((1 / ((x1 * x2) * x0)) * (((1 / x0) + (1 / x2)) + (1 / x1))) - ((1 / ((x2 * x1) * x3)) * (((1 / x3) + (1 / x1)) + (1 / x2))))) / (x0 - x3)
( x0 <> x1 & x0 <> x2 & x1 <> x2 ) by A3, ZFMISC_1:def 6;
then x0,x1,x2 are_mutually_distinct by ZFMISC_1:def 5;
then A4: [!f,x0,x1,x2!] = (k / ((x0 * x1) * x2)) * (((1 / x0) + (1 / x1)) + (1 / x2)) by A1, A2, DIFF_3:49
.= (k * (1 / ((x1 * x2) * x0))) * (((1 / x0) + (1 / x2)) + (1 / x1)) ;
( x1 <> x2 & x1 <> x3 & x2 <> x3 ) by A3, ZFMISC_1:def 6;
then x1,x2,x3 are_mutually_distinct by ZFMISC_1:def 5;
then [!f,x1,x2,x3!] = (k / ((x1 * x2) * x3)) * (((1 / x1) + (1 / x2)) + (1 / x3)) by A1, A2, DIFF_3:49;
hence [!f,x0,x1,x2,x3!] = (k * (((1 / ((x1 * x2) * x0)) * (((1 / x0) + (1 / x2)) + (1 / x1))) - ((1 / ((x2 * x1) * x3)) * (((1 / x3) + (1 / x1)) + (1 / x2))))) / (x0 - x3) by A4; :: thesis: verum