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