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 Th78:
  (A\andB)\andC\iffA\and(B\andC) in F
  proof
A1: A\andB\andC\impA\imp(A\andB\andC\impB\andC\imp((A\andB)\andC\imp
    (A\and(B\andC)))) in F by Th49;
    A\andB\impA in F & A\andB\andC\impA\andB in F by Def38; then
    A\andB\andC\impA in F by Th45; then
A2: A\andB\andC\impB\andC\imp(A\andB\andC\impA\and(B\andC)) in F by A1,Def38;
    A\andB\impB in F & C\impC in F by Def38,Th34; then
    A\andB\andC\impB\andC in F by Th72;
    then
A3: (A\andB)\andC\impA\and(B\andC) in F by A2,Def38;
A4: (A\and(B\andC))\impA\andB\imp((A\and(B\andC))\impC\imp(A\and(B\andC)\imp
    (A\andB\andC))) in F by Th49;
    B\andC\impB in F & A\impA in F by Def38,Th34; then
    A\and(B\andC)\imp(A\andB) in F by Th72; then
A5: (A\and(B\andC))\impC\imp(A\and(B\andC)\imp (A\andB\andC)) in F by A4,Def38;
    B\andC\impC in F & A\and(B\andC)\impB\andC in F by Def38; then
    A\and(B\andC)\impC in F by Th45;
    then
    (A\and(B\andC)\imp (A\andB\andC)) in F by A5,Def38;
    hence thesis by A3,Th43;
  end;
