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 Th20:
  rng x_r-seq(m) c= ].0,PI/2.[
  proof
    set f = x_r-seq(m);
    let y be object;
    assume y in rng f;
    then consider n such that
A1: 1 <= n and
A2: n <= len f and
A3: y = f.n by Lm2;
A4: len f = m by Th19;
    then
A5: f.n = n*PI/(2*m+1) by A1,A2,Th19;
    2*n <= 2*m by A2,A4,XREAL_1:64;
    then 2*n+1 <= 2*m+1 by XREAL_1:6;
    then
A6: n*PI/(2*n+1) >= n*PI/(2*m+1) by XREAL_1:118;
A7: ((2*n+1)*n")" = (2*n+1)"*n" " by XCMPLX_1:204;
A8: 2*n/n = 2 by A1,XCMPLX_1:89;
    2+1/n > 2+0 by A1,XREAL_1:8;
    then ((2*n+1)*n")" < (2*1")" by A8,XREAL_1:88;
    then PI*(n/(2*n+1)) < PI*(1/2) by A7,XREAL_1:68;
    then n*PI/(2*m+1) < PI/2 by A6,XXREAL_0:2;
    hence thesis by A1,A3,A5,XXREAL_1:4;
  end;
