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
  1 <= k <= m implies 2/(k*PI) + (x_r-seq(m))".k = (x_r-seq(m+1))".k
  proof
    assume that
A1: 1 <= k and
A2: k <= m;
    set f = x_r-seq(m);
    set g = x_r-seq(m+1);
    m+0 <= m+1 by XREAL_1:6;
    then k <= m+1 by A2,XXREAL_0:2;
    then
A3: g.k = k*PI/(2*(m+1)+1) by A1,Th19;
    f.k = k*PI/(2*m+1) by A1,A2,Th19;
    then (2*m+1)/(k*PI) = (f.k)" by XCMPLX_1:213
    .= f".k by VALUED_1:10;
    hence 2/(k*PI) + f".k = (2*(m+1)+1)/(k*PI)
    .= (g.k)" by A3,XCMPLX_1:213
    .= g".k by VALUED_1:10;
  end;
