 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
  a => (b "/\" c) = (a => b) "/\" (a => c)
  proof
A1: -((a => b) "/\" (a => c)) < -(a => (b "/\" c))
    proof
A2:   -(a => b) < (a "/\" (-b)) by Def10;
      (-b) < (-b) "\/" (-c) by Th7; then
      (-b) < -(b "/\" c) by Th8; then
A3:   a "/\" (-b) < a "/\" -(b "/\" c) by Lm1;
A4:   -(a => b) < a "/\" -(b "/\" c) by A2,A3,Def3;
      a "/\" -(b "/\" c) < -(a => (b "/\" c)) by Def9; then
A5:   -(a => b) < -(a => (b "/\" c)) by A4,Def3;
A6:   -(a => c) < (a "/\" (-c)) by Def10;
      (-c) < (-b) "\/" (-c) by Th7; then
      (-c) < -(b "/\" c) by Th8; then
      a "/\" (-c) < a "/\" -(b "/\" c) by Lm1; then
A7:   -(a => c) < a "/\" -(b "/\" c) by A6,Def3;
      a "/\" -(b "/\" c) < -(a => (b "/\" c)) by Def9; then
      -(a => c) < -(a => (b "/\" c)) by A7,Def3; then
      (-(a => b)) "\/" (-(a => c)) < -(a => (b "/\" c)) by A5,Def7;
      hence thesis by Th8;
    end;
A8: -(a => (b "/\" c)) < -((a => b) "/\" (a => c))
    proof
A9:   -(a => (b "/\" c)) < a "/\" -(b "/\" c) by Def10;
A10:   a "/\" (-b) < -(a => b) by Def9;
      a "/\" (-c) < -(a => c) by Def9; then
      (a "/\" (-b)) "\/" (a "/\" (-c)) < (-(a => b)) "\/" (-(a => c))
        by A10,Lm2; then
      a "/\" ((-b) "\/" (-c)) < (-(a => b)) "\/" (-(a => c))
        by LATTICES:def 11; then
      a "/\" -(b "/\" c) < (-(a => b)) "\/" (-(a => c)) by Th8; then
      a "/\" -(b "/\" c) < -((a => b) "/\" (a => c)) by Th8;
      hence thesis by A9,Def3;
    end;
A11: a => (b "/\" c) < (a => b) "/\" (a => c)
    proof
A12:  a => (b "/\" c) < a => b by Th6,Th16bis;
      a => (b "/\" c) < a => c by Th6,Th16bis;
      hence thesis by A12,Def8;
    end;
A13: (a => b) "/\" (a => c) < a => (b "/\" c)
    proof
      a => b < a => b by Def2; then
A14:  a "/\" (a => b) < b by Def4;
      a => c < a => c by Def2; then
      a "/\" (a => c) < c by Def4; then
      (a "/\" (a => b)) "/\" (a "/\" (a => c)) < b "/\" c by A14,Lm2; then
      (a "/\" (a => b)) "/\"  a "/\" (a => c) < b "/\" c
        by LATTICES:def 7; then
      (a "/\" a) "/\" (a => b) "/\" (a => c) < b "/\" c
        by LATTICES:def 7; then
      a "/\" ((a => b) "/\" (a => c)) < b "/\" c by LATTICES:def 7;
      hence thesis by Def4;
    end;
A15: a => (b "/\" c) <= (a => b) "/\" (a => c) by A1,A11,Th5;
    (a => b) "/\" (a => c) <= a => (b "/\" c) by A8,A13,Th5;
    hence thesis by A15,Th3;
  end;
