reserve h,r,r1,r2,x0,x1,x2,x3,x4,x5,x,a,b,c,k for Real,
  f,f1,f2 for Function of REAL,REAL;

theorem Th36:
  (for x holds f.x = k/x) & x0<>0 & x1<>0 & x2<>0 & x3<>0 & x0,x1,
  x2,x3 are_mutually_distinct implies [!f,x0,x1,x2,x3!]=-k/(x0*x1*x2*x3)
proof
  assume that
A1: for x holds f.x = k/x and
A2: x0<>0 and
A3: x1<>0 & x2<>0 and
A4: x3<>0;
  assume
A5: x0,x1,x2,x3 are_mutually_distinct;
  then
A6: x1<>x2 by ZFMISC_1:def 6;
  x1<>x3 & x2<>x3 by A5,ZFMISC_1:def 6;
  then
A7: x1,x2,x3 are_mutually_distinct by A6,ZFMISC_1:def 5;
A8: x0-x3<>0 by A5,ZFMISC_1:def 6;
  x0<>x1 & x0<>x2 by A5,ZFMISC_1:def 6;
  then x0,x1,x2 are_mutually_distinct by A6,ZFMISC_1:def 5;
  then [!f,x0,x1,x2,x3!] = (k/(x0*x1*x2)-[!f,x1,x2,x3!])/(x0-x3) by A1,A2,A3
,Th35
    .= (k/(x0*x1*x2)-k/(x1*x2*x3))/(x0-x3) by A1,A3,A4,A7,Th35
    .= (k*x3/(x0*x1*x2*x3)-k/(x1*x2*x3)) /(x0-x3) by A4,XCMPLX_1:91
    .= (k*x3/(x0*x1*x2*x3)-k*x0/(x0*(x1*x2*x3))) /(x0-x3) by A2,XCMPLX_1:91
    .= ((k*x3-k*x0)/(x0*x1*x2*x3))/(x0-x3) by XCMPLX_1:120
    .= (-k)*(x0-x3)/((x0*x1*x2*x3)*(x0-x3)) by XCMPLX_1:78
    .= (-k)/(x0*x1*x2*x3) by A8,XCMPLX_1:91;
  hence thesis by XCMPLX_1:187;
end;
