theorem
  FinJoin B1 "\/" FinJoin B2 = FinJoin (B1 \/ B2)
proof
  set J = the L_join of 0L;
  thus FinJoin (B1 \/ B2) = J $$ (B1 \/ B2,id 0L) by LATTICE2:def 3
    .= J $$ (B1,id 0L) "\/" J $$ (B2,id 0L) by SETWISEO:33
    .= FinJoin B1 "\/" J $$ (B2,id 0L) by LATTICE2:def 3
    .=FinJoin B1 "\/" FinJoin B2 by LATTICE2:def 3;
end;
