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;

theorem
  x0,x1,x2 are_mutually_distinct implies [!f,x0,x1,x2!] = [!f,x0,x2,x1!]
proof
  assume x0,x1,x2 are_mutually_distinct;
  then
A1: x2-x1<>0 & x2-x0<>0 & x1-x0<>0 by ZFMISC_1:def 5;
  set x10 = x1-x0;
  set x20 = x2-x0;
  set x21 = x2-x1;
A2: [!f,x0,x2,x1!] = ((f.x0-f.x2)/(-(x2-x0)) - (f.x2-f.x1)/(x2-x1))/(-(x1-x0))
    .= (-(f.x0-f.x2)/(x2-x0) - (f.x2-f.x1)/(x2-x1))/(-(x1-x0)) by XCMPLX_1:188
    .= (-((f.x0-f.x2)/(x2-x0) + (f.x2-f.x1)/(x2-x1)))/(-(x1-x0))
    .= ((f.x0-f.x2)/(x2-x0) + (f.x2-f.x1)/(x2-x1))/(x1-x0) by XCMPLX_1:191
    .= (((f.x0-f.x2)*x21 + (f.x2-f.x1)*x20)/(x20*x21))/x10 by A1,XCMPLX_1:116
    .= (f.x0*x21 - f.x1*x20 + f.x2*x10)/(x20*x21*x10) by XCMPLX_1:78;
   [!f,x0,x1,x2!] = ((f.x0-f.x1)/(-(x1-x0))
                     - (f.x1-f.x2)/(-(x2-x1)))/(-(x2-x0))
    .= (-(f.x0-f.x1)/(x1-x0) - (f.x1-f.x2)/(-(x2-x1)))/(-(x2-x0))
                                                            by XCMPLX_1:188
    .= (-(f.x0-f.x1)/(x1-x0) - (-(f.x1-f.x2)/(x2-x1)))/(-(x2-x0))
                                                            by XCMPLX_1:188
    .= (-((f.x0-f.x1)/(x1-x0) - (f.x1-f.x2)/(x2-x1)))/(-(x2-x0))
    .= ((f.x0-f.x1)/(x1-x0) - (f.x1-f.x2)/(x2-x1))/(x2-x0) by XCMPLX_1:191
    .= (((f.x0-f.x1)*x21 - (f.x1-f.x2)*x10)/(x10*x21))/x20 by A1,XCMPLX_1:130
    .= (f.x0*x21 - f.x1*x20 + f.x2*x10)/(x10*x21*x20) by XCMPLX_1:78
    .= [!f,x0,x2,x1!] by A2;
hence thesis;
end;
