reserve L for Lattice;
reserve X,Y,Z,V for Element of L;
reserve L for D_Lattice;
reserve X,Y,Z for Element of L;
reserve L for 0_Lattice;
reserve X,Y,Z for Element of L;
reserve L for B_Lattice;
reserve X,Y,Z,V for Element of L;

theorem Th49:
  X meets Y "\/" Z iff X meets Y or X meets Z
proof
  thus X meets Y "\/" Z implies X meets Y or X meets Z
  proof
    assume X meets Y "\/" Z;
    then X "/\" (Y "\/" Z) <> Bottom L;
    then (X "/\" Y) "\/" (X "/\" Z) <> Bottom L by LATTICES:def 11;
    then (X "/\" Y) <> Bottom L or (X "/\" Z) <> Bottom L;
    hence thesis;
  end;
  assume
A1: X meets Y or X meets Z;
  per cases by A1;
  suppose
A2: X meets Y;
    X "/\" Y [= (X "/\" Y) "\/" (X "/\" Z) by LATTICES:5;
    then
A3: X "/\" Y [= X "/\" (Y "\/" Z) by LATTICES:def 11;
    X "/\" (Y "\/" Z) <> Bottom L by A3,Th9,A2;
   hence thesis;
  end;
  suppose
A4: X meets Z;
A5: (X "/\" Z) "\/" (X "/\" Y) = X "/\" (Y "\/" Z) by LATTICES:def 11;
    X "/\" Z <> Bottom L by A4;
    then X "/\" (Y "\/" Z) <> Bottom L by A5,Th11;
    hence thesis;
  end;
end;
