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 Th34:
  (for x holds f.x = k/x) & x0<>x1 & x0<>0 & x1<>0 implies [!f,x0,
  x1!] = - k/(x0*x1)
proof
  assume that
A1: for x holds f.x = k/x and
A2: x0<>x1 and
A3: x0<>0 and
A4: x1<>0;
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 thesis by A5,XCMPLX_1:89;
end;
