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\or(B\andC) \imp (A\orB)\and(A\orC) in F
  proof
    set AB =A\orB, AC = A\orC, BC = B\andC;
    set ABC = A\orBC;
A1: ABC\impAB\imp(ABC\impAC\imp(ABC\impAB\andAC)) in F by Th49;
A2: A\impAB\imp(BC\impAB\imp(ABC\impAB)) in F by Def38;
A3: A\impAC\imp(BC\impAC\imp(ABC\impAC)) in F by Def38;
A4: A\impAC in F & A\impAB in F by Def38;
    BC\impC in F & BC\impB in F & B\impAB in F & C\impAC in F by Def38; then
A5: BC\impAB in F & BC\impAC in F by Th45;
    BC\impAB\imp(ABC\impAB) in F & BC\impAC\imp(ABC\impAC) in F by A2,A3,A4,
Def38;
    then
A6: ABC\impAB in F & ABC\impAC in F by A5,Def38; then
    ABC\impAC\imp(ABC\impAB\andAC) in F by A1,Def38;
    hence thesis by A6,Def38;
  end;
