theorem Th63:
  for m being non zero Nat ex n being Nat st for n1,n2 being Nat st
    n <= n1 & n <= n2 holds |. dblseq_ex_1.(n1,n2) - 0 .| < 1/m
  proof
    let m be non zero Nat;
    now
      let n1,n2 be Nat;
      assume that
A1:   m <= n1 and
      m <= n2;
      m + 0 < n1 + 1 by A1,XREAL_1:8;
      then 1/(n1 + 1) < 1/m by XREAL_1:76;
      then |.1/(n1 + 1) - 0.| < 1/m by ABSVALUE:def 1;
      hence |.dblseq_ex_1.(n1,n2) - 0.| < 1/m by Def5;
    end;
    hence ex n being Nat st for n1,n2 being Nat st
    n <= n1 & n <= n2 holds |. dblseq_ex_1.(n1,n2) - 0 .| < 1/m;
  end;
