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 Th81:
  A\and(B\orC) \imp (A\andB)\or(A\andC) in F
  proof
    set AB =A\andB, AC = A\andC, BC = B\orC;
    set ABC = A\andBC;
A1: (\notA\or\notB)\and(\notA\or\notC) \imp \notA\or(\notB\and\notC) in F
    by Th80;
    \not(A\andB)\imp(\notA\or\notB) in F &
    \not(A\andC)\imp(\notA\or\notC) in F by Th70; then
A2: \not(A\andB)\and\not(A\andC)\imp(\notA\or\notB)\and(\notA\or\notC) in F
    by Th72;
    \not(A\andB\orA\andC)\imp\not(A\andB)\and\not(A\andC) in F by Th71; then
    \not(A\andB\orA\andC)\imp(\notA\or\notB)\and(\notA\or\notC) in F
    by A2,Th45; then
A3: \not(A\andB\orA\andC)\imp\notA\or(\notB\and\notC) in F by A1,Th45;
    \notA\imp\notA in F &
    \notB\and\notC\imp\not(B\orC) in F by Th34,Th74; then
    \notA\or(\notB\and\notC)\imp\notA\or\not(B\orC) in F &
    \notA\or\not(B\orC)\imp\not(A\and(B\orC)) in F
    by Th73,Th59; then
    \notA\or(\notB\and\notC)\imp\not(A\and(B\orC)) in F
    by Th45; then
    \not(A\andB\orA\andC)\imp\not(A\and(B\orC)) in F by A3,Th45;
    hence thesis by Th58;
  end;
