reserve x,y,X,X1,Y,Z for set;
reserve L for Lattice;
reserve F,H for Filter of L;
reserve p,q,r for Element of L;
reserve L1, L2 for Lattice;
reserve a1,b1 for Element of L1;
reserve a2 for Element of L2;
reserve f for Homomorphism of L1,L2;
reserve B for Element of Fin the carrier of L;
reserve DL for distributive Lattice;
reserve f for Homomorphism of DL,L2;
reserve 0L for lower-bounded Lattice,
  B,B1,B2 for Element of Fin the carrier of 0L,
  b for Element of 0L;
reserve f for UnOp of the carrier of 0L;

theorem Th13:
  FinJoin(B \/ {.b.},f) = FinJoin (B,f) "\/" f.b
proof
  set J= the L_join of 0L;
  thus FinJoin (B \/ {.b.},f) = J $$ (B \/ {.b.},f) by LATTICE2:def 3
    .= J $$ (B,f) "\/" f.b by SETWISEO:32
    .= FinJoin (B,f) "\/" f.b by LATTICE2:def 3;
end;
