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 Th19:
  len (x_r-seq m) = m &
  for k st 1 <= k <= m holds x_r-seq(m).k = k*PI/(2*m+1)
proof
  A1: dom (x_r-seq m) = dom (idseq m) by VALUED_1:def 5;
  hence len (x_r-seq m)=m by FINSEQ_3:29;
  let k;assume 1 <= k <= m;
  then k in dom (x_r-seq m) & k in Seg m by A1,FINSEQ_3:25;
  then (x_r-seq m).k = (PI/(2*m+1)) * ((idseq m).k) & (idseq m).k = k
        by VALUED_1:def 5,FINSEQ_2:49;
  hence thesis;
end;
