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,x3 are_mutually_distinct implies
  [!f,x0,x1,x2,x3!] = 0
proof
  assume that
A1:for x holds f.x = x^2 and
A2:x0,x1,x2,x3 are_mutually_distinct;
A3:f.x0 = x0^2 & f.x1 = x1^2 & f.x2 = x2^2 & f.x3 = x3^2 by A1;
A4:x0-x1<>0 & x1-x2<>0 & x2-x3<>0 & x0-x2<>0 & x1-x3<>0 & x0-x3<>0
                                                     by A2,ZFMISC_1:def 6;
  [!f,x0,x1,x2,x3!] = ((((x0-x1)*(x0+x1))/(x0-x1)-((x1-x2)*(x1+x2))
       /(x1-x2))/(x0-x2)-(((x1-x2)*(x1+x2))/(x1-x2)-((x2-x3)*(x2+x3))
       /(x2-x3))/(x1-x3))/(x0-x3) by A3
    .= (((x0+x1)-((x1-x2)*(x1+x2))/(x1-x2))/(x0-x2)
       -(((x1-x2)*(x1+x2))/(x1-x2)-((x2-x3)*(x2+x3))/(x2-x3))/(x1-x3))/(x0-x3)
                                                         by A4,XCMPLX_1:89
    .= (((x0+x1)-(x1+x2))/(x0-x2)
       -(((x1-x2)*(x1+x2))/(x1-x2)-((x2-x3)*(x2+x3))/(x2-x3))/(x1-x3))/(x0-x3)
                                                         by A4,XCMPLX_1:89
    .= (((x0+x1)-(x1+x2))/(x0-x2)
       -((x1+x2)-((x2-x3)*(x2+x3))/(x2-x3))/(x1-x3))/(x0-x3)
                                                         by A4,XCMPLX_1:89
    .= (((x0+x1)-(x1+x2))/(x0-x2)-((x1+x2)-(x2+x3))/(x1-x3))/(x0-x3)
                                                       by A4,XCMPLX_1:89
    .= (1-(x1-x3)/(x1-x3))/(x0-x3) by A4,XCMPLX_1:60
    .= (1-1)/(x0-x3) by A4,XCMPLX_1:60
    .= 0;
  hence thesis;
end;
