theorem
  L is distributive & x [= y [= z implies x [= z
  proof
    assume
A0: L is distributive;
    assume
A1: x [= y [= z;
    x "/\" z = x "/\" (y "\/" z) by A1,LATTICES:def 3
            .= (x "/\" y) "\/" (x "/\" z) by A0,LATTICES:def 11
            .= x "\/" (x "/\" z) by A1,LATTICES:4
            .= x by LATTICES:def 8;
    hence thesis by LATTICES:4;
  end;
