let x0, x1, k be Real; :: thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = k / x ) & x0 <> x1 & x0 <> 0 & x1 <> 0 holds
[!f,x0,x1!] = - (k / (x0 * x1))
let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = k / x ) & x0 <> x1 & x0 <> 0 & x1 <> 0 implies [!f,x0,x1!] = - (k / (x0 * x1)) )
assume that
A1: for x being Real holds f . x = k / x and
A2: x0 <> x1 and
A3: x0 <> 0 and
A4: x1 <> 0 ; :: thesis: [!f,x0,x1!] = - (k / (x0 * x1))
A5: x1 - x0 <> 0 by A2;
[!f,x0,x1!] = ((k / x0) - (f . x1)) / (x0 - x1) by A1
.= ((k / x0) - (k / x1)) / (x0 - x1) by A1
.= (((k * x1) / (x0 * x1)) - (k / x1)) / (x0 - x1) by A4, XCMPLX_1:91
.= (((k * x1) / (x0 * x1)) - ((k * x0) / (x0 * x1))) / (x0 - x1) by A3, XCMPLX_1:91
.= (((k * x1) - (k * x0)) / (x0 * x1)) / (x0 - x1) by XCMPLX_1:120
.= ((k * (x1 - x0)) / (x0 * x1)) / (- (x1 - x0))
.= - (((k * (x1 - x0)) / (x0 * x1)) / (x1 - x0)) by XCMPLX_1:188
.= - (((k * (x1 - x0)) / (x1 - x0)) / (x0 * x1)) by XCMPLX_1:48 ;
hence [!f,x0,x1!] = - (k / (x0 * x1)) by A5, XCMPLX_1:89; :: thesis: verum