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
  x0,x1,x2 are_mutually_distinct implies [!f,x0,x1,x2!] = f.x0/((x0-x1)
  *(x0-x2))+f.x1/((x1-x0)*(x1-x2))+f.x2/((x2-x0)*(x2-x1))
proof
  assume
A1: x0,x1,x2 are_mutually_distinct;
  then
A2: x1-x2<>0 by ZFMISC_1:def 5;
A3: x0-x1<>0 by A1,ZFMISC_1:def 5;
A4: x0-x2<>0 by A1,ZFMISC_1:def 5;
  [!f,x0,x1,x2!] = (f.x0-f.x1)/(x0-x1)/(x0-x2) - (f.x1-f.x2)/(x1-x2)/(x0-
  x2) by XCMPLX_1:120
    .= (f.x0-f.x1)/((x0-x1)*(x0-x2)) - (f.x1-f.x2)/(x1-x2)/(x0-x2) by
XCMPLX_1:78
    .= (f.x0-f.x1)/((x0-x1)*(x0-x2)) - (f.x1-f.x2)/((x1-x2)*(x0-x2)) by
XCMPLX_1:78
    .= f.x0/((x0-x1)*(x0-x2)) - f.x1/((x0-x1)*(x0-x2)) - (f.x1-f.x2)/((x1-x2
  )*(x0-x2)) by XCMPLX_1:120
    .= f.x0/((x0-x1)*(x0-x2)) - f.x1/((x0-x1)*(x0-x2)) - (f.x1/((x1-x2)*(x0-
  x2)) - f.x2/((x1-x2)*(x0-x2))) by XCMPLX_1:120
    .= f.x0/((x0-x1)*(x0-x2))-(f.x1/((x0-x1)*(x0-x2)) + f.x1/((x1-x2)*(x0-x2
  )))+ f.x2/((x1-x2)*(x0-x2))
    .= f.x0/((x0-x1)*(x0-x2)) - (f.x1*(x1-x2)/((x0-x1) *(x0-x2)*(x1-x2)) + f
  .x1/((x1-x2)*(x0-x2))) + f.x2/((x1-x2)*(x0-x2)) by A2,XCMPLX_1:91
    .= f.x0/((x0-x1)*(x0-x2)) - (f.x1*(x1-x2)/((x0-x1)*(x0-x2) *(x1-x2)) + f
.x1*(x0-x1)/((x1-x2)*(x0-x2)*(x0-x1))) + f.x2/((x1-x2)*(x0-x2)) by A3,
XCMPLX_1:91
    .= f.x0/((x0-x1)*(x0-x2)) - (f.x1*(x1-x2) + f.x1*(x0-x1))/((x0-x1)*(x0-
  x2)*(x1-x2)) + f.x2/((x1-x2)*(x0-x2)) by XCMPLX_1:62
    .= f.x0/((x0-x1)*(x0-x2)) - f.x1*(x0-x2)/((x0-x1) *(x1-x2)*(x0-x2)) + f.
  x2/((x1-x2)*(x0-x2))
    .= f.x0/((x0-x1)*(x0-x2)) - f.x1/(-(x1-x0)*(x1-x2)) + f.x2/(-(x2-x1)*(x0
  -x2)) by A4,XCMPLX_1:91
    .= f.x0/((x0-x1)*(x0-x2)) + - f.x1/(-((x1-x0)*(x1-x2))) + f.x2/((-(x2-x1
  ))*(-(x2-x0)));
  hence thesis by XCMPLX_1:189;
end;
