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
  x0,x1,x2 are_mutually_distinct implies
  [!f,x0,x1,x2!] = [!f,x2,x0,x1!] & [!f,x0,x1,x2!] = [!f,x1,x0,x2!]
proof
  assume
A1: x0,x1,x2 are_mutually_distinct;
  then
A2: x1<>x2 by ZFMISC_1:def 5;
  x0<>x1 & x0<>x2 by A1,ZFMISC_1:def 5;
  then
A3: x2,x0,x1 are_mutually_distinct by A2,ZFMISC_1:def 5;
  then [!f,x0,x1,x2!] = [!f,x2,x0,x1!] by Th34;
  hence thesis by A3,Th34;
end;
