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
  (for x 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)
proof
  assume that
A1: for x holds f.x = k/x and
A2: x0<>0 and
A3: x1<>0 & x2<>0 & x3<>0 and
A4: x4<>0;
  assume
A5: x0,x1,x2,x3,x4 are_mutually_distinct;
  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 thesis by A9,XCMPLX_1:91;
end;
