reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;

theorem
  (for n holds F.n = Y /\ less_eq_dom(f,r-1/(n+1))) implies Y /\
  less_dom(f,r) = union rng F
proof
  assume for n holds F.n = Y /\ less_eq_dom(f,r-1/(n+1));
  then
  for n be Element of NAT holds F.n =Y /\ less_eq_dom(R_EAL f,(r-1/(n
  +1)));
  then Y /\ less_dom(f,r) = union rng F by MESFUNC1:21;
  hence thesis;
end;
