theorem Th29:
  (ex x st x in B & u [= f.x) implies u [= FinJoin(B,f)
proof
  given x such that
A1: x in B and
A2: u [= f.x;
  f.x [= FinJoin(B,f) by A1,Th28;
  then
A3: f.x "\/" FinJoin(B,f) = FinJoin(B,f);
  then u "\/" FinJoin(B,f) = u "\/" f.x "\/" FinJoin(B,f) by LATTICES:def 5
    .= FinJoin(B,f) by A2,A3;
  hence thesis;
end;
