 reserve L for Quasi-Boolean_Algebra,
         x, y, z for Element of L;
 reserve L for Nelson_Algebra,
         a, b, c, d, x, y, z for Element of L;

theorem Th12:
  (a => b) "/\" ((-a) "\/" b) = (-a) "\/" b
  proof
A1: -((-a) "\/" b) < -((a => b) "/\" ((-a) "\/" b))
    proof
A2:   -((-a) "\/" b) = (--a) "/\" (-b) by Th1;
A3:   (--a) "/\" (-b) = a "/\" (-b) by ROBBINS3:def 6;
      a "/\" (-b) < -(a => b) by Def9; then
      (-((-a) "\/" b)) "\/" (-((-a) "\/" b)) <
        (-(a => b)) "\/" (-((-a) "\/" b)) by A2,A3,Lm1;
      hence thesis by Th8;
    end;
A4: ((-a) "\/" b) <= ((a => b) "/\" ((-a) "\/" b))
    proof
      ((-a) "\/" b) <= (a => b) by Th11;
      hence thesis;
    end;
    (a => b) "/\" ((-a) "\/" b) <= (-a) "\/" b by Th6,A1,Th5;
    hence thesis by A4,Th3;
  end;
