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 Th54:
  (A\andB)\or(A\andC) \imp A\and(B\orC) in F
  proof
    A\andB\impB in F & B\impB\orC in F by Def38; then
A1: A\andB\impB\orC in F by Th45;
    A\andC\impC in F & C\impB\orC in F by Def38; then
A2: A\andC\impB\orC in F by Th45;
    set AB = A\andB, AC = A\andC;
A3: AB\impA\imp(AB\impB\orC\imp(AB\impA\and(B\orC))) in F by Th49;
    AB\impA in F by Def38; then
    AB\impB\orC\imp(AB\impA\and(B\orC)) in F by A3,Def38; then
A4: AB\impA\and(B\orC) in F by A1,Def38;
A5: AC\impA\imp(AC\impB\orC\imp(AC\impA\and(B\orC))) in F by Th49;
    AC\impA in F by Def38; then
    AC\impB\orC\imp(AC\impA\and(B\orC)) in F by A5,Def38; then
A6: AC\impA\and(B\orC) in F by A2,Def38;
    AB\impA\and(B\orC)\imp(AC\impA\and(B\orC)\imp(AB\orAC\impA\and(B\orC)))in F
    by Def38; then
    AC\impA\and(B\orC)\imp(AB\orAC\impA\and(B\orC)) in F by A4,Def38;
    hence thesis by A6,Def38;
  end;
