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,x3 are_mutually_distinct implies [!f,x0,x1,x2,x3!] = [!f,x1,
  x2,x3,x0!] & [!f,x0,x1,x2,x3!] = [!f,x3,x2,x1,x0!]
proof
  set x10 = x1-x0;
  set x20 = x2-x0;
  set x30 = x3-x0;
  set x12 = x1-x2;
  set x13 = x1-x3;
  set x23 = x2-x3;
  assume
A1: x0,x1,x2,x3 are_mutually_distinct;
  then
A2: x2-x3<>0 by ZFMISC_1:def 6;
A3: x3-x0<>0 by A1,ZFMISC_1:def 6;
A4: x1-x3<>0 by A1,ZFMISC_1:def 6;
A5: x1-x0<>0 by A1,ZFMISC_1:def 6;
A6: x2-x0<>0 by A1,ZFMISC_1:def 6;
A7: x1-x2<>0 by A1,ZFMISC_1:def 6;
A8: [!f,x0,x1,x2,x3!] = (((f.x0-f.x1)/(-(x1-x0))-(f.x1-f.x2)/(x1-x2)) /(-(x2
  -x0))-((f.x1-f.x2)/(x1-x2)-(f.x2-f.x3) /(x2-x3))/(x1-x3))/(x0-x3)
    .= ((-(f.x0-f.x1)/(x1-x0)-(f.x1-f.x2)/(x1-x2)) /(-(x2-x0))-((f.x1-f.x2)/
  (x1-x2)-(f.x2-f.x3) /(x2-x3))/(x1-x3))/(x0-x3) by XCMPLX_1:188
    .= ((-((f.x0-f.x1)/(x1-x0)+(f.x1-f.x2)/(x1-x2))) /(-(x2-x0))-((f.x1-f.x2
  )/(x1-x2)-(f.x2-f.x3) /(x2-x3))/(x1-x3))/(x0-x3)
    .= (((f.x0-f.x1)/(x1-x0)+(f.x1-f.x2)/(x1-x2))/(x2-x0) -((f.x1-f.x2)/(x1-
  x2)-(f.x2-f.x3) /(x2-x3))/(x1-x3))/(x0-x3) by XCMPLX_1:191
    .= (((f.x0-f.x1)*x12+(f.x1-f.x2)*x10) /(x10*x12)/x20-((f.x1-f.x2)/(x1-x2
  ) -(f.x2-f.x3)/(x2-x3))/(x1-x3))/(x0-x3) by A7,A5,XCMPLX_1:116
    .= (((f.x0-f.x1)*x12+(f.x1-f.x2)*x10)/(x10*x12)/x20 -((f.x1-f.x2)*x23-(f
  .x2-f.x3)*x12) /(x12*x23)/x13)/(x0-x3) by A2,A7,XCMPLX_1:130
    .= (((f.x0-f.x1)*x12+(f.x1-f.x2)*x10)/(x10*x12*x20) -((f.x1-f.x2)*x23-(f
  .x2-f.x3)*x12) /(x12*x23)/x13)/(x0-x3) by XCMPLX_1:78
    .= (((f.x0-f.x1)*x12+(f.x1-f.x2)*x10)/(x10*x12*x20) -((f.x1-f.x2)*x23-(f
  .x2-f.x3)*x12) /(x12*x23*x13))/(x0-x3) by XCMPLX_1:78
    .= (-(((f.x0-f.x1)*x12+(f.x1-f.x2)*x10)/(x10*x12*x20) -((f.x1-f.x2)*x23-
  (f.x2-f.x3)*x12) /(x12*x23*x13)))/(-(x0-x3)) by XCMPLX_1:191
    .= (((f.x1-f.x2)*x23-(f.x2-f.x3)*x12)/(x12*x23*x13) -((f.x0-f.x1)*x12+(f
  .x1-f.x2)*x10)/(x10*x12*x20))/(x3-x0)
    .= ((f.x1-f.x2)*x23-(f.x2-f.x3)*x12)/(x12*x23*x13)/x30 -((f.x0-f.x1)*x12
  +(f.x1-f.x2)*x10) /(x10*x12*x20)/x30 by XCMPLX_1:120
    .= ((f.x1-f.x2)*x23-(f.x2-f.x3)*x12)/(x12*x23*x13*x30) -((f.x0-f.x1)*x12
  +(f.x1-f.x2)*x10) /(x10*x12*x20)/x30 by XCMPLX_1:78
    .= ((f.x1-f.x2)*x23-(f.x2-f.x3)*x12)/(x12*x23*x13*x30) -((f.x0-f.x1)*x12
  +(f.x1-f.x2)*x10) /(x10*x12*x20*x30) by XCMPLX_1:78
    .= ((f.x1-f.x2)*x23-(f.x2-f.x3)*x12)*x20 /(x12*x23*x13*x30*x20)-((f.x0-f
  .x1)*x12+(f.x1-f.x2)*x10) /(x10*x12*x20*x30) by A6,XCMPLX_1:91
    .= ((f.x1-f.x2)*x23*x20-(f.x2-f.x3)*x12*x20)*x10 /(x12*x20*x30*x23*x13*
  x10)-((f.x0-f.x1)*x12 +(f.x1-f.x2)*x10)/(x10*x12*x20*x30) by A5,XCMPLX_1:91
    .= ((f.x1-f.x2)*x23*x20*x10-(f.x2-f.x3)*x12*x20*x10) /(x12*x20*x30*x23*
x13*x10)-((f.x0-f.x1)*x12+(f.x1-f.x2)*x10) *(x23*x13)/((x12*x20*x30)*x10*(x23*
  x13)) by A2,A4,XCMPLX_1:6,91
    .= (((f.x1-f.x2)*x23*x20*x10-(f.x2-f.x3)*x12*x20*x10) -((f.x0-f.x1)*x12*
  x23*x13+(f.x1-f.x2)*x10*x23*x13)) /(x12*x20*x30*x23*x13*x10) by XCMPLX_1:120
    .= (-f.x0*x12*x23*x13+f.x1*x20*x23*x30-f.x2*x13*x10*x30 +f.x3*x12*x20*
  x10)/(x12*x20*x30*x23*x13*x10);
A9: [!f,x3,x2,x1,x0!] = (((f.x3-f.x2)/(-(x2-x3))-(f.x2-f.x1)/(-(x1-x2))) /(
  -(x1-x3))-((f.x2-f.x1)/(-(x1-x2))-(f.x1-f.x0) /(x1-x0))/(x2-x0))/(x3-x0)
    .= ((-(f.x3-f.x2)/(x2-x3)-(f.x2-f.x1)/(-(x1-x2))) /(-(x1-x3))-((f.x2-f.
  x1)/(-(x1-x2))-(f.x1-f.x0) /(x1-x0))/(x2-x0))/(x3-x0) by XCMPLX_1:188
    .= ((-(f.x3-f.x2)/(x2-x3)-(-(f.x2-f.x1)/(x1-x2))) /(-(x1-x3))-((f.x2-f.
  x1)/(-(x1-x2))-(f.x1-f.x0) /(x1-x0))/(x2-x0))/(x3-x0) by XCMPLX_1:188
    .= ((-((f.x3-f.x2)/(x2-x3)-(f.x2-f.x1)/(x1-x2))) /(-(x1-x3))-((f.x2-f.x1
  )/(-(x1-x2))-(f.x1-f.x0) /(x1-x0))/(x2-x0))/(x3-x0)
    .= (((f.x3-f.x2)/(x2-x3)-(f.x2-f.x1)/(x1-x2))/(x1-x3) -((f.x2-f.x1)/(-(
  x1-x2))-(f.x1-f.x0)/(x1-x0)) /(x2-x0))/(x3-x0) by XCMPLX_1:191
    .= (((f.x3-f.x2)/(x2-x3)-(f.x2-f.x1)/(x1-x2))/(x1-x3) -(-(f.x2-f.x1)/(x1
  -x2)-(f.x1-f.x0)/(x1-x0)) /(x2-x0))/(x3-x0) by XCMPLX_1:188
    .= (((f.x3-f.x2)/x23-(f.x2-f.x1)/x12)/x13+ -((-((f.x1-f.x0)/x10+(f.x2-f.
  x1)/x12)))/x20)/x30
    .= (((f.x3-f.x2)/x23-(f.x2-f.x1)/x12)/x13+((f.x1-f.x0)/x10 +(f.x2-f.x1)/
  x12)/x20)/x30 by XCMPLX_1:190
    .= (((f.x3-f.x2)*x12-(f.x2-f.x1)*x23)/(x23*x12)/x13 +((f.x1-f.x0)/x10+(f
  .x2-f.x1) /x12)/x20)/x30 by A2,A7,XCMPLX_1:130
    .= (((f.x3-f.x2)*x12-(f.x2-f.x1)*x23)/(x23*x12)/x13 +((f.x1-f.x0)*x12+(f
  .x2-f.x1)*x10) /(x10*x12)/x20)/x30 by A7,A5,XCMPLX_1:116
    .= (((f.x3-f.x2)*x12-(f.x2-f.x1)*x23)/(x23*x12*x13) +((f.x1-f.x0)*x12+(f
  .x2-f.x1)*x10) /(x10*x12)/x20)/x30 by XCMPLX_1:78
    .= (((f.x3-f.x2)*x12-(f.x2-f.x1)*x23)/(x23*x12*x13) +((f.x1-f.x0)*x12+(f
  .x2-f.x1)*x10) /(x10*x12*x20))/x30 by XCMPLX_1:78
    .= (((f.x3-f.x2)*x12-(f.x2-f.x1)*x23)/(x23*x12*x13) +((f.x1-f.x0)*x12+(f
  .x2-f.x1)*x10)*x13 /(x10*x12*x20*x13))/x30 by A4,XCMPLX_1:91
    .= (((f.x3-f.x2)*x12-(f.x2-f.x1)*x23)/(x23*x12*x13) +((f.x1-f.x0)*x12*
  x13+(f.x2-f.x1)*x10*x13)*x23 /(x10*x12*x13*x20*x23))/x30 by A2,XCMPLX_1:91
    .= (((f.x3-f.x2)*x12-(f.x2-f.x1)*x23)*x20/(x23*x12*x13*x20) +((f.x1-f.x0
  )*x12*x13*x23+(f.x2-f.x1)*x10*x13*x23)/ (x10*x12*x13*x20*x23))/x30 by A6,
XCMPLX_1:91
    .= (((f.x3-f.x2)*x12*x20-(f.x2-f.x1)*x23*x20)*x10/(x12*x13*x20*x23*x10)
+((f.x1-f.x0)*x12*x13*x23+(f.x2-f.x1)*x10*x13*x23)/ (x10*x12*x13*x20*x23))/x30
  by A5,XCMPLX_1:91
    .= ((f.x3-f.x2)*x12*x20*x10-(f.x2-f.x1)*x23*x20*x10) /(x10*x12*x13*x20*
  x23)/x30+((f.x1-f.x0)*x12*x13*x23 +(f.x2-f.x1)*x10*x13*x23)/(x10*x12*x13*x20*
  x23)/x30 by XCMPLX_1:62
    .= ((f.x3-f.x2)*x12*x20*x10-(f.x2-f.x1)*x23*x20*x10) /(x10*x12*x13*x20*
  x23*x30)+((f.x1-f.x0)*x12*x13*x23 +(f.x2-f.x1)*x10*x13*x23)/(x10*x12*x13*x20*
  x23) /x30 by XCMPLX_1:78
    .= ((f.x3-f.x2)*x12*x20*x10-(f.x2-f.x1)*x23*x20*x10) /(x10*x12*x13*x20*
x23*x30)+((f.x1-f.x0)*x12*x13*x23 +(f.x2-f.x1)*x10*x13*x23)/ (x10*x12*x13*x20*
  x23*x30) by XCMPLX_1:78
    .= ((f.x3-f.x2)*x12*x20*x10-(f.x2-f.x1)*x23*x20*x10 +((f.x1-f.x0)*x12*
  x13*x23+(f.x2-f.x1)*x10*x13*x23)) /(x10*x12*x13*x20*x23*x30) by XCMPLX_1:62
    .= [!f,x0,x1,x2,x3!] by A8;
  [!f,x1,x2,x3,x0!] = (((f.x1-f.x2)*x23-(f.x2-f.x3)*x12)/(x12*x23)/x13 -((
f.x2-f.x3)/(x2-x3)-(f.x3-f.x0)/(x3-x0)) /(x2-x0))/(x1-x0) by A2,A7,XCMPLX_1:130
    .= (((f.x1-f.x2)*x23-(f.x2-f.x3)*x12)/(x12*x23)/x13- ((f.x2-f.x3)*x30-(f
  .x3-f.x0)*x23) /(x23*x30)/x20)/x10 by A2,A3,XCMPLX_1:130
    .= (((f.x1-f.x2)*x23-(f.x2-f.x3)*x12)/(x12*x23*x13) -((f.x2-f.x3)*x30-(f
  .x3-f.x0)*x23) /(x23*x30)/x20)/x10 by XCMPLX_1:78
    .= (((f.x1-f.x2)*x23-(f.x2-f.x3)*x12)/(x12*x23*x13) -((f.x2-f.x3)*x30-(f
  .x3-f.x0)*x23) /(x23*x30*x20))/x10 by XCMPLX_1:78
    .= ((f.x1-f.x2)*x23-(f.x2-f.x3)*x12)/(x12*x23*x13) /x10-((f.x2-f.x3)*x30
  -(f.x3-f.x0)*x23) /(x23*x30*x20)/x10 by XCMPLX_1:120
    .= ((f.x1-f.x2)*x23-(f.x2-f.x3)*x12)/(x12*x23*x13*x10) -((f.x2-f.x3)*x30
  -(f.x3-f.x0)*x23) /(x23*x30*x20)/x10 by XCMPLX_1:78
    .= ((f.x1-f.x2)*x23-(f.x2-f.x3)*x12)/(x12*x23*x13*x10) -((f.x2-f.x3)*x30
  -(f.x3-f.x0)*x23) /(x23*x30*x20*x10) by XCMPLX_1:78
    .= ((f.x1-f.x2)*x23-(f.x2-f.x3)*x12)*x30 /(x12*x23*x13*x10*x30)-((f.x2-f
  .x3)*x30 -(f.x3-f.x0)*x23)/(x23*x30*x20*x10) by A3,XCMPLX_1:91
    .= ((f.x1-f.x2)*x23*x30-(f.x2-f.x3)*x12*x30)*x20 /(x12*x23*x13*x10*x30*
  x20)-((f.x2-f.x3)*x30 -(f.x3-f.x0)*x23)/(x23*x30*x20*x10) by A6,XCMPLX_1:91
    .=((f.x1-f.x2)*x23*x30*x20-(f.x2-f.x3)*x12*x30*x20) /(x12*x23*x13*x10*
x30*x20)-((f.x2-f.x3)*x30 -(f.x3-f.x0)*x23)*x13/ ((x10*x30*x20)*x23*x13) by A4,
XCMPLX_1:91
    .=((f.x1-f.x2)*x23*x30*x20 -(f.x2-f.x3)*x12*x30*x20)/(x12*x23*x13*x10*
x30*x20) -((f.x2-f.x3)*x30*x13-(f.x3-f.x0)*x23*x13)*x12 /((x23*x13*x10*x30*x20)
  *x12) by A7,XCMPLX_1:91
    .=(((f.x1-f.x2)*x23*x30*x20-(f.x2-f.x3)*x12*x30*x20)- ((f.x2-f.x3)*x30*
  x13*x12-(f.x3-f.x0)*x23*x13*x12)) /(x12*x23*x13*x10*x30*x20) by XCMPLX_1:120
    .= [!f,x0,x1,x2,x3!] by A8;
  hence thesis by A9;
end;
