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;
reserve DL for distributive upper-bounded Lattice,
  B for Element of Fin the carrier of DL,
  p for Element of DL,
  f for UnOp of the carrier of DL;
reserve CL for C_Lattice;
reserve IL for implicative Lattice;
reserve f for Homomorphism of IL,CL;
reserve i,j,k for Element of IL;

theorem Th27:
  f is one-to-one implies (f.i "/\" f.k [= f.j implies f.k [= f.(i => j))
proof
  assume
A1: f is one-to-one;
  hereby
    assume f.i "/\" f.k [= f.j;
    then f.(i "/\" k) [= f.j by D2;
    then i "/\" k [= j by A1,Th5;
    then k [= (i => j) by FILTER_0:def 7;
    hence f.k [= f.(i => j) by Th4;
  end;
end;
