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 Th76:
  (A\orB)\orC\iffA\or(B\orC) in F
  proof
A1: A\orB\imp(A\or(B\orC))\imp(C\imp(A\or(B\orC))\imp((A\orB)\orC\imp
    (A\or(B\orC)))) in F by Def38;
    B\impB\orC in F & A\impA in F by Def38,Th34; then
    A\orB\imp(A\or(B\orC)) in F by Th59; then
A2: C\impA\or(B\orC)\imp(A\orB\orC\impA\or(B\orC)) in F by A1,Def38;
    C\impB\orC in F & B\orC\impA\or(B\orC) in F by Def38; then
    C\impA\or(B\orC) in F by Th45;
    then
A3: (A\orB)\orC\impA\or(B\orC) in F by A2,Def38;
A4: A\imp((A\orB)\orC)\imp(B\orC\imp(A\orB\orC)\imp(A\or(B\orC)\imp
    (A\orB\orC))) in F by Def38;
    B\impA\orB in F & C\impC in F by Def38,Th34; then
    B\orC\imp(A\orB\orC) in F by Th59; then
A5: A\impA\orB\orC\imp(A\or(B\orC)\impA\orB\orC) in F by A4,Th46;
    A\impA\orB in F & A\orB\impA\orB\orC in F by Def38; then
    A\impA\orB\orC in F by Th45;
    then
    A\or(B\orC)\impA\orB\orC in F by A5,Def38;
    hence thesis by A3,Th43;
  end;
