reserve X,Y for set, x,y,z for object, i,j,n for natural number;
reserve
  n for non empty Nat,
  S for non empty non void n PC-correct PCLangSignature,
  L for language MSAlgebra over S,
  F for PC-theory of L,
  A,B,C,D for Formula of L;

theorem
  A\iffB\imp(A\or\notB)\and(\notA\orB) in F
  proof
    \notA\and(A\or\notB) \imp (A\or\notB)\and\notA in F &
    B\and(A\or\notB) \imp (A\or\notB)\andB in F by Th50; then
    (A\or\notB)\and\notA\or(A\or\notB)\andB\imp(A\or\notB)\and(\notA\orB) in F&
    \notA\and(A\or\notB)\orB\and(A\or\notB) \imp
    (A\or\notB)\and\notA\or(A\or\notB)\andB in F by Th54,Th59; then
A1: \notA\and(A\or\notB)\orB\and(A\or\notB)\imp(A\or\notB)\and(\notA\orB) in F
    by Th45;
    (\notA\andA)\or(\notA\and\notB)\imp\notA\and(A\or\notB) in F &
    (B\andA)\or(B\and\notB)\impB\and(A\or\notB) in F &
    (\notA\and\notB)\imp(\notA\andA)\or(\notA\and\notB) in F &
    (B\andA)\imp(B\andA)\or(B\and\notB) in F by Def38,Th54; then
    (\notA\and\notB)\imp\notA\and(A\or\notB) in F &
    (B\andA)\impB\and(A\or\notB) in F by Th45; then
    (\notA\and\notB)\or(B\andA)\imp\notA\and(A\or\notB)\orB\and(A\or\notB) in F
    by Th59; then
A2: (\notA\and\notB)\or(B\andA)\imp(A\or\notB)\and(\notA\orB) in F by A1,Th45;
    A\andB\impB\andA in F & \notA\and\notB\imp\notA\and\notB in F
    by Th34,Th50; then
    (\notA\and\notB)\or(A\andB)\imp(\notA\and\notB)\or(B\andA) in F &
    (A\andB)\or(\notA\and\notB)\imp(\notA\and\notB)\or(A\andB) in F
    by Th36,Th59; then
    A\andB\or\notA\and\notB\imp(\notA\and\notB)\or(B\andA) in F by Th45; then
A3: A\andB\or\notA\and\notB\imp(A\or\notB)\and(\notA\orB) in F by A2,Th45;
    A\iffB\impA\andB\or\notA\and\notB in F by Th86;
    hence thesis by A3,Th45;
  end;
