reserve X for set,
  x,y,z for Element of BooleLatt X,
  s for set;
reserve y for Element of BooleLatt X;
reserve L for Lattice,
  p,q for Element of L;
reserve A for RelStr,
  a,b,c for Element of A;
reserve A for non empty RelStr,
  a,b,c,c9 for Element of A;
reserve V for with_suprema antisymmetric RelStr,
  u1,u2,u3,u4 for Element of V;
reserve N for with_infima antisymmetric RelStr,
  n1,n2,n3,n4 for Element of N;
reserve K for with_suprema with_infima reflexive antisymmetric RelStr,
  k1,k2,k3 for Element of K;

theorem Th16:
  N is transitive implies (n1 "/\" n2) "/\" n3 = n1 "/\" (n2 "/\" n3)
proof
  assume
A1: N is transitive;
A2: n1"/\"n2 <= n1 by Lm2;
A3: n1"/\"n2 <= n2 by Lm2;
A4: n2"/\"n3 <= n2 by Lm2;
A5: n2"/\"n3 <= n3 by Lm2;
A6: (n1"/\"n2)"/\"n3 <= n1"/\"n2 by Lm2;
A7: (n1"/\"n2)"/\"n3 <= n3 by Lm2;
A8: (n1"/\"n2)"/\"n3 <= n1 by A1,A2,A6,ORDERS_2:3;
  (n1"/\"n2)"/\"n3 <= n2 by A1,A3,A6,ORDERS_2:3;
  then
A9: (n1"/\"n2)"/\"n3 <= n2"/\"n3 by A7,Lm2;
  now
    let n4;
    assume that
A10: n4 <= n1 and
A11: n4 <= n2"/\"n3;
A12: n4 <= n2 by A1,A4,A11,ORDERS_2:3;
A13: n4 <= n3 by A1,A5,A11,ORDERS_2:3;
    n4 <= n1"/\"n2 by A10,A12,Lm2;
    hence n4 <= (n1"/\"n2)"/\"n3 by A13,Lm2;
  end;
  hence thesis by A8,A9,Def14;
end;
