reserve n,m for Element of NAT;
reserve h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
reserve f,f1,f2 for Function of REAL,REAL;
reserve S for Seq_Sequence;

theorem
  (for x holds f.x = x^2) & x0,x1,x2 are_mutually_distinct implies
  [!f,x0,x1,x2!] = 1
proof
  assume that
A1:for x holds f.x = x^2 and
A2:x0,x1,x2 are_mutually_distinct;
A3:f.x0 = x0^2 & f.x1 = x1^2 & f.x2 = x2^2 by A1;
A4:x0-x1<>0 & x1-x2<>0 & x0-x2<>0 by A2,ZFMISC_1:def 5;
  [!f,x0,x1,x2!] = (((x0-x1)*(x0+x1))/(x0-x1)
       -((x1-x2)*(x1+x2))/(x1-x2))/(x0-x2) by A3
    .= ((x0+x1)-((x1-x2)*(x1+x2))/(x1-x2))/(x0-x2) by A4,XCMPLX_1:89
    .= ((x0+x1)-(x1+x2))/(x0-x2) by A4,XCMPLX_1:89
    .= 1 by A4,XCMPLX_1:60;
  hence thesis;
end;
