 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 Th14:
  a "/\" (a => b) = a "/\" ((-a) "\/" b)
  proof
A1: a "/\" (a => b) < a "/\" ((-a) "\/" b)
    proof
      a => b < a => b by Def2; then
A2:   a "/\" (a => b) < b by Def4;
      b < b "\/" (-a) by Th7; then
A3:   a "/\" (a => b) < (-a) "\/" b by A2,Def3;
      a "/\" (a => b) < a by Th6;
      hence thesis by A3,Def8;
    end;
A4: -(a "/\" ((-a)"\/"b)) < -(a "/\" (a => b))
    proof
A5:   -(a "/\" ((-a) "\/" b)) = (-a) "\/" -((-a) "\/" b) by Th8;
A6:   (-a) "\/" -((-a) "\/" b) = (-a) "\/" ((--a) "/\" (-b)) by Th1;
A7:   (-a) "\/" ((--a) "/\" (-b)) = (-a) "\/" (a "/\" (-b))
        by ROBBINS3:def 6;
A8:   (a "/\" (-b)) < -(a => b) by Def9;
      -(a => b) < (-a) "\/" -(a => b) by Th7; then
A9:   -(a => b) < -(a "/\" (a => b)) by Th8;
A10:  (-a) < (-a) "\/" -(a => b) by Th7;
A11:  (a "/\" (-b)) < -(a "/\" (a => b)) by A8,A9,Def3;
      (-a) < -(a "/\" (a => b)) by A10,Th8;
      hence thesis by A5,A6,A7,A11,Def7;
    end;
A12: a "/\" ((-a) "\/" b) < a "/\" (a => b)
    proof
      ((-a) "\/" b) < (a => b) by Th13;
      hence thesis by Lm1;
    end;
A13: -(a "/\" (a => b)) < -(a "/\" ((-a)"\/"b))
    proof
      -(a => b) < a "/\" (-b) by Def10; then
      (-a) "\/" -(a => b) < (-a) "\/" (a "/\" (-b)) by Lm1; then
      -(a "/\" (a => b)) < (-a) "\/" (a "/\" (-b)) by Th8; then
      -(a "/\" (a => b)) < (-a) "\/" ((--a) "/\" (-b))
        by ROBBINS3:def 6; then
      -(a "/\" (a => b)) < (-a) "\/" -((-a) "\/" b) by Th1;
      hence thesis by Th8;
    end;
A14: a "/\" ((-a)"\/"b) <= a "/\" (a => b) by A12,A13,Th5;
    a "/\" (a => b) <= a "/\" ((-a) "\/" b) by A1,A4,Th5;
    hence thesis by A14,Th3;
  end;
