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;
reserve 1L for upper-bounded Lattice,
  B,B1,B2 for Element of Fin the carrier of 1L,
  b for Element of 1L;
reserve f,g for UnOp of the carrier of 1L;

theorem Th20:
  FinMeet(B \/ {.b.},f) = FinMeet (B,f) "/\" f.b
proof
  set M= the L_meet of 1L;
  thus FinMeet (B \/ {.b.},f) = M $$ (B \/ {.b.},f) by LATTICE2:def 4
    .= M $$ (B,f) "/\" f.b by SETWISEO:32
    .= FinMeet (B,f) "/\" f.b by LATTICE2:def 4;
end;
