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 Th82:
  A\impB\imp\notA\orB in F
  proof
A1: A\impB\imp(A\impB) in F by Th34;
    A\impB\imp(A\impB)\imp((A\impB)\andA\impB) in F by Th48; then
    (A\impB)\andA\impB in F & \notA\imp\notA in F by A1,Def38,Th34; then
A2: \notA\or(A\impB)\andA\imp\notA\orB in F by Th59;
    (\notA\or(A\impB))\and(\notA\orA)\imp(\notA\or(A\impB)\andA) in F by Th80;
    then
A3: (\notA\or(A\impB))\and(\notA\orA)\imp\notA\orB in F by A2,Th45;
    (\notA\orA)\and(\notA\or(A\impB))\imp((\notA\or(A\impB))\and(\notA\orA))
    in F by Th50; then
A4: (\notA\orA)\and(\notA\or(A\impB))\imp\notA\orB in F by A3,Th45;
    ((\notA\orA)\and(\notA\or(A\impB))\imp\notA\orB)\imp
    ((\notA\orA)\imp((\notA\or(A\impB))\imp\notA\orB)) in F by Th47; then
A5: (\notA\orA)\imp((\notA\or(A\impB))\imp\notA\orB) in F by A4,Def38;
    A\or\notA in F & A\or\notA\imp\notA\orA in F by Def38,Th36; then
    \notA\orA in F by Def38; then
A6: (\notA\or(A\impB))\imp\notA\orB in F by A5,Def38;
    (A\impB)\imp(\notA\or(A\impB)) in F by Def38;
    hence thesis by A6,Th45;
  end;
