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 Th49:
  ( C\impA )\imp(( C\impB )\imp( C\imp( A\andB ))) in F
  proof
A1: ( C\imp( B\imp( A\andB )))\imp(( C\impB )\imp( C\imp( A\andB ))) in F
    by Def38;
    A\imp( B\imp( A\andB )) in F by Def38;
    then
A2: C\imp( A\imp( B\imp( A\andB ))) in F by Th44;
    (C\imp( A\imp( B\imp( A\andB ))))\imp(( C\impA )\imp( C\imp( B\imp( A
    \andB )))) in F by Def38;
    then ( C\impA )\imp( C\imp( B\imp( A\andB ))) in F by A2,Def38;
    hence thesis by A1,Th45;
  end;
