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 Th13:
  u1 "\/" u2 = u2 "\/" u1
proof
A1: u1 <= u1"\/"u2 by Lm1;
A2: u2 <= u1"\/"u2 by Lm1;
  for u3 st u2 <= u3 & u1 <= u3 holds u1"\/"u2 <= u3 by Lm1;
  hence thesis by A1,A2,Def13;
end;
