
theorem Th20:
  for f being U-continuous Function st dom f is subset-closed for
  a being set st a in dom f holds f.a = union (f.:Fin a)
proof
  let f be U-continuous Function such that
A1: dom f is subset-closed;
  let a be set;
  assume
A2: a in dom f;
  then reconsider
  C = dom f as d.union-closed subset-closed non empty set by A1;
  reconsider a as Element of C by A2;
  f.a = f.union Fin a by Th19;
  hence thesis by Def10;
end;
