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 Th32:
  B <> {} & (for x st x in B holds f.x [= u) implies FinJoin(B,f) [= u
proof
  assume that
A1: B <> {} and
A2: for x st x in B holds f.x [= u;
  set J = the L_join of L;
A3: now
    let x;
    assume x in B;
    then
A4: f.x [= u by A2;
    thus (J[:](f,u)).x = f.x "\/" u by FUNCOP_1:48
      .= u by A4;
  end;
  FinJoin(B,f) "\/" u = J$$(B,J[:](f,u)) by A1,Th20,SETWISEO:28
    .= u by A1,A3,SETWISEO:24;
  hence thesis;
end;
