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;

theorem Th11:
  f is onto implies L2 is distributive
proof
  assume
A1: f is onto;
  thus L2 is distributive
  proof
    let a,b,c be Element of L2;
    consider a9 be Element of DL such that
A2: f.a9=a by A1,Th6;
    consider c9 be Element of DL such that
A3: f.c9=c by A1,Th6;
    consider b9 be Element of DL such that
A4: f.b9=b by A1,Th6;
    thus a"/\"(b"\/"c) = a"/\"f.(b9"\/"c9) by A4,A3,D1
      .= f.(a9"/\"(b9"\/"c9)) by A2,D2
      .=f.((a9"/\"b9)"\/"(a9"/\"c9)) by LATTICES:def 11
      .=f.(a9"/\"b9)"\/"f.(a9"/\"c9) by D1
      .=(a"/\"b)"\/"f.(a9"/\"c9) by A2,A4,D2
      .=(a"/\"b)"\/"(a"/\"c) by A2,A3,D2;
  end;
end;
