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 Th21:
  1 <= k <= m implies 0 < x_r-seq(m).k < PI/2
  proof
    set f = x_r-seq(m);
A1: rng f c= ].0,PI/2.[ by Th20;
A2: len f = m by Th19;
    assume 1 <= k <= m;
    then k in dom f by A2,FINSEQ_3:25;
    then f.k in rng f by FUNCT_1:def 3;
    hence thesis by A1,XXREAL_1:4;
  end;
