theorem Th19:
  f|A is bounded & i in dom D implies upper_bound rng(f|divset(D,i
  )) >= lower_bound rng f
proof
  assume
A1: f|A is bounded;
  assume i in dom D;
  then divset(D,i) c= A by INTEGRA1:8;
  hence thesis by A1,Lm4;
end;
