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