reserve k,m,n for Nat;
reserve R for commutative Ring,
        p,q for Polynomial of R,
        z0,z1 for Element of R;

theorem Th31:
  Sum(Basel-seq,m) <= Basel-seq2.m
  proof
    set 2m1=2*m+1,2m2=2*m+2;
A1: (2*m*2m2/6) /(2m1^2)/(PI^2)"
       = ((2*m*2m2) / (2m1*2m1)) *(PI^2) *6"
      .= (2*m/2m1) *(2m2 /2m1)  *(PI^2) *6" by XCMPLX_1:76
      .= (PI^2/6)*(2*m/2m1) *(2m2 /2m1)
      .= Basel-seq2.m by BASEL_1:33;
    Sum sqr cosec x_r-seq(m) >= Sum ((sqr x_r-seq(m))") by BASEL_1:30;
    then 2*m*2m2/6 >= Sum ((sqr x_r-seq(m))") by Th29;
    then 2*m*2m2/6 >= 2m1^2 / (PI^2) * Sum(Basel-seq,m) by BASEL_1:35;
    then 2*m*2m2/6 >= 2m1^2 * ((PI^2)" * Sum(Basel-seq,m));
    then (2*m*2m2/6) /(2m1^2) >= (PI^2)" * Sum(Basel-seq,m) by XREAL_1:77;
    hence thesis by A1,XREAL_1:77;
  end;
