reserve A for set,
  C for non empty set,
  B for Subset of A,
  x for Element of A,
  f,g for Function of A,C;
reserve B for Element of Fin A;
reserve L for non empty LattStr,
  a,b,c for Element of L;
reserve L for Lattice;
reserve a,b,c,u,v for Element of L;
reserve A for non empty set,
  x for Element of A,
  B for Element of Fin A,
  f,g for Function of A, the carrier of L;

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;
