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 Th86:
  A\iffB\impA\andB\or\notA\and\notB in F
  proof
A1: (A\iffB)\imp(A\impB) in F & (A\iffB)\imp(B\impA) in F by Def38;
    A\impB\imp\notA\orB in F & B\impA\imp\notB\orA in F by Th82; then
A2: (A\iffB)\imp\notA\orB in F & (A\iffB)\imp\notB\orA in F by A1,Th45;
    (A\iffB)\imp\notA\orB\imp((A\iffB)\imp\notB\orA\imp((A\iffB)\imp
    (\notA\orB)\and(\notB\orA))) in F by Th49; then
    (A\iffB)\imp\notB\orA\imp((A\iffB)\imp(\notA\orB)\and(\notB\orA)) in F
    by A2,Def38; then
A3: (A\iffB)\imp(\notA\orB)\and(\notB\orA) in F by A2,Def38;
A4: (\notA\orB)\and(\notB\orA) \imp (\notA\orB)\and\notB\or(\notA\orB)\andA
    in F by Th81;
    (\notA\orB)\and\notB \imp \notB\and(\notA\orB) in F &
    \notB\and(\notA\orB)\imp\notB\and\notA\or\notB\andB in F by Th50,Th81;then
    (\notA\orB)\and\notB \imp \notB\and\notA\or\notB\andB in F &
    \notB\and\notA\or\notB\andB\imp\notB\and\notA in F by Th45,Th84; then
A5: (\notA\orB)\and\notB \imp \notB\and\notA in F by Th45;
    (\notA\orB)\andA \imp A\and(\notA\orB) in F &
    A\and(\notA\orB)\impA\and\notA\orA\andB in F by Th50,Th81;then
    (\notA\orB)\andA \imp A\and\notA\orA\andB in F &
    A\and\notA\orA\andB \imp A\andB\orA\and\notA in F by Th36,Th45; then
    (\notA\orB)\andA \imp A\andB\orA\and\notA in F &
    A\andB\orA\and\notA\impA\andB in F by Th45,Th85; then
    (\notA\orB)\andA \imp A\andB in F by Th45; then
    (\notA\orB)\and\notB\or(\notA\orB)\andA \imp \notB\and\notA\orA\andB in F
    by A5,Th59; then
    (\notA\orB)\and(\notB\orA) \imp \notB\and\notA\orA\andB in F by A4,Th45;
    then
A6: (A\iffB)\imp\notB\and\notA\orA\andB in F by A3,Th45;
    \notB\and\notA\imp\notA\and\notB in F & A\andB\impA\andB in F by Th34,Th50;
    then
    \notB\and\notA\orA\andB \imp \notA\and\notB\orA\andB in F &
    \notA\and\notB\orA\andB \imp A\andB\or\notA\and\notB in F by Th36,Th59;then
    \notB\and\notA\orA\andB \imp A\andB\or\notA\and\notB in F by Th45;
    hence thesis by A6,Th45;
  end;
