reserve X for set;
reserve k,m,n for Nat;
reserve i for Integer;
reserve a,b,c,d,e,g,p,r,x,y for Real;
reserve z for Complex;

theorem Th25:
  x_r-seq(n) is one-to-one
proof
  set f= x_r-seq(n);
  let x1,x2 be object such that
A1: x1 in dom f & x2 in dom f & f.x1=f.x2;
  reconsider x1,x2 as Nat by A1;
  len f=n by Th19;then
  1 <= x1 <= n & 1<= x2 <= n by A1,FINSEQ_3:25;
  then
A2: f.x1= x1*PI/(2*n+1) & f.x2 =  x2 *PI/(2*n+1) by Th19;
  x1*PI=x2 *PI by A1,A2,XCMPLX_1:53;
  hence thesis by XCMPLX_1:5;
end;
