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 Th14:
  V is transitive implies (u1 "\/" u2) "\/" u3 = u1 "\/" (u2 "\/" u3)
proof
  assume
A1: V is transitive;
A2: u1 <= u1"\/"u2 by Lm1;
A3: u2 <= u1"\/"u2 by Lm1;
A4: u2 <= u2"\/"u3 by Lm1;
A5: u3 <= u2"\/"u3 by Lm1;
A6: u1"\/"u2 <= (u1"\/"u2)"\/"u3 by Lm1;
A7: u3 <= (u1"\/"u2)"\/"u3 by Lm1;
A8: u1 <= (u1"\/"u2)"\/"u3 by A1,A2,A6,ORDERS_2:3;
  u2 <= (u1"\/"u2)"\/"u3 by A1,A3,A6,ORDERS_2:3;
  then
A9: u2"\/"u3 <= (u1"\/"u2)"\/"u3 by A7,Lm1;
  now
    let u4;
    assume that
A10: u1 <= u4 and
A11: u2"\/"u3 <= u4;
A12: u2 <= u4 by A1,A4,A11,ORDERS_2:3;
A13: u3 <= u4 by A1,A5,A11,ORDERS_2:3;
    u1"\/"u2 <= u4 by A10,A12,Lm1;
    hence (u1"\/"u2)"\/"u3 <= u4 by A13,Lm1;
  end;
  hence thesis by A8,A9,Def13;
end;
