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 Th35:
  (for x holds f.x = k/x) & x0<>0 & x1<>0 & x2<>0 & x0,x1,x2
  are_mutually_distinct implies [!f,x0,x1,x2!] = k/(x0*x1*x2)
proof
  assume that
A1: for x holds f.x = k/x and
A2: x0<>0 and
A3: x1<>0 and
A4: x2<>0;
  assume
A5: x0,x1,x2 are_mutually_distinct;
  then
A6: x1<>x2 by ZFMISC_1:def 5;
A7: x0-x2<>0 by A5,ZFMISC_1:def 5;
  x0<>x1 by A5,ZFMISC_1:def 5;
  then [!f,x0,x1,x2!] = (-k/(x0*x1)-[!f,x1,x2!])/(x0-x2) by A1,A2,A3,Th34
    .= (-k/(x0*x1)- -k/(x1*x2))/(x0-x2) by A1,A3,A4,A6,Th34
    .= (-k/(x0*x1)+k/(x1*x2))/(x0-x2)
    .= (-k*x2/(x0*x1*x2)+k/(x1*x2))/(x0-x2) by A4,XCMPLX_1:91
    .= (-k*x2/(x0*x1*x2)+k*x0/(x0*(x1*x2))) /(x0-x2) by A2,XCMPLX_1:91
    .= (-(k*x2/(x0*x1*x2)-k*x0/(x0*x1*x2)))/(x0-x2)
    .= (-(k*x2-k*x0)/(x0*x1*x2))/(x0-x2) by XCMPLX_1:120
    .= (-k*(x2-x0))/(x0*x1*x2)/(x0-x2) by XCMPLX_1:187
    .= k*(x0-x2)/((x0*x1*x2)*(x0-x2)) by XCMPLX_1:78;
  hence thesis by A7,XCMPLX_1:91;
end;
