 reserve n,m,i,p for Nat,
         h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
 reserve f,f1,f2,g for Function of REAL,REAL;

theorem
  (for x holds f.x = k/(x^2)) & x0<>0 & x1<>0 & x2<>0 & x3<>0
  & x0,x1,x2,x3 are_mutually_distinct implies
  [!f,x0,x1,x2,x3!] = k*((1/(x1*x2*x0))*(1/x0+1/x2+1/x1)
  -(1/(x2*x1*x3))*(1/x3+1/x1+1/x2))/(x0-x3)
proof
  assume that
A1:for x holds f.x = k/(x^2) and
A2:x0<>0 & x1<>0 & x2<>0 & x3<>0 and
A3: x0,x1,x2,x3 are_mutually_distinct;
  x0 <> x1 & x0 <> x2 & x1 <> x2 by A3,ZFMISC_1:def 6;
  then x0,x1,x2 are_mutually_distinct by ZFMISC_1:def 5; then
A4:  [!f,x0,x1,x2!] = (k/(x0*x1*x2))*(1/x0+1/x1+1/x2) by A1,A2,DIFF_3:49
          .= k*(1/(x1*x2*x0))*(1/x0+1/x2+1/x1);
  x1 <> x2 & x1 <> x3 & x2 <> x3 by A3,ZFMISC_1:def 6;
  then x1,x2,x3 are_mutually_distinct by ZFMISC_1:def 5;
  then [!f,x1,x2,x3!] = (k/(x1*x2*x3))*(1/x1+1/x2+1/x3) by A1,A2,DIFF_3:49;
  hence thesis by A4;
end;
