reserve n,m,k,i for Nat,
  h,r,r1,r2,x0,x1,x2,x for Real,
  S for Functional_Sequence of REAL,REAL,
  y for set;
reserve f,f1,f2 for Function of REAL,REAL;

theorem Th34:
  x0,x1,x2 are_mutually_distinct implies [!f,x0,x1,x2!] = [!f,x1,
  x2,x0!] & [!f,x0,x1,x2!] = [!f,x2,x1,x0!]
proof
  set x10 = x1-x0;
  set x20 = x2-x0;
  set x12 = x1-x2;
  assume
A1: x0,x1,x2 are_mutually_distinct;
  then
A2: x1-x2<>0 by ZFMISC_1:def 5;
A3: x1-x0<>0 by A1,ZFMISC_1:def 5;
A4: [!f,x0,x1,x2!] = ((f.x0-f.x1)/(-(x1-x0)) - (f.x1-f.x2)/(x1-x2))/(-(x2-x0 ))
    .= (-(f.x0-f.x1)/(x1-x0) - (f.x1-f.x2)/(x1-x2))/(-(x2-x0)) by XCMPLX_1:188
    .= (-((f.x0-f.x1)/(x1-x0) + (f.x1-f.x2)/(x1-x2)))/(-(x2-x0))
    .= ((f.x0-f.x1)/(x1-x0) + (f.x1-f.x2)/(x1-x2))/(x2-x0) by XCMPLX_1:191
    .= (((f.x0-f.x1)*x12 + (f.x1-f.x2)*x10)/(x10*x12))/x20 by A2,A3,
XCMPLX_1:116
    .= (f.x0*x12 + f.x1*x20 - f.x2*x10)/(x10*x12*x20) by XCMPLX_1:78;
A5: [!f,x2,x1,x0!] = ((-(f.x1-f.x2))/(-(x1-x2)) - (f.x1-f.x0)/(x1-x0))/(x2- x0)
    .= ((f.x1-f.x2)/(x1-x2) - (f.x1-f.x0)/(x1-x0))/(x2-x0) by XCMPLX_1:191
    .= (((f.x1-f.x2)*x10 - (f.x1-f.x0)*x12)/(x12*x10))/x20 by A2,A3,
XCMPLX_1:130
    .= [!f,x0,x1,x2!] by A4,XCMPLX_1:78;
  x2-x0<>0 by A1,ZFMISC_1:def 5;
  then
  [!f,x1,x2,x0!] = (((f.x1-f.x2)*x20 - (f.x2-f.x0)*x12)/(x12*x20))/x10 by A2,
XCMPLX_1:130
    .= (f.x0*x12 + f.x1*x20 - f.x2*x10)/(x12*x20*x10) by XCMPLX_1:78
    .= [!f,x0,x1,x2!] by A4;
  hence thesis by A5;
end;
