 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 Th10:
  b <= c implies a "\/" b <= a "\/" c & a "/\" b <= a "/\" c
  proof
    assume
A1: b <= c;
A2: a "\/" b < a "\/" c
    proof
      b < c by A1,Th5;
      hence thesis by Lm1;
    end;
A3: -(a "\/" c) < -(a "\/" b)
    proof
      -c < - b by A1,Th5; then
      (-a) "/\" (-c) < (-a) "/\" (-b) by Lm1; then
      -(a "\/" c) < (-a) "/\" (-b) by Th1;
      hence thesis by Th1;
    end;
A4: a "/\" b < a "/\" c
    proof
      b < c by A1,Th5;
      hence thesis by Lm1;
    end;
    -(a "/\" c) < -(a "/\" b)
    proof
      -c < -b by A1,Th5; then
      (-a) "\/" (-c) < (-a) "\/" (-b) by Lm1; then
      -(a "/\" c) < (-a) "\/" (-b) by Th8;
      hence thesis by Th8;
    end;
    hence thesis by A4,Th5,A2,A3;
  end;
