theorem Th45:
  seq is bounded_above implies ((for m st n<=m holds seq.m <= r)
  iff (superior_realsequence seq).n <= r)
proof
  assume
A1: seq is bounded_above;
  thus (for m st n<=m holds seq.m <= r) implies (superior_realsequence seq).n
  <= r
  proof
    assume for m st n<=m holds seq.m <= r;
    then for k holds seq.(n+k) <= r by NAT_1:11;
    hence thesis by A1,Th44;
  end;
  assume
A2: (superior_realsequence seq).n <= r;
  now
    let m;
    assume n<=m;
    then consider k being Nat such that
A3: m = n + k by NAT_1:10;
    thus seq.m <= r by A1,A2,A3,Th44;
  end;
  hence thesis;
end;
