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 Th47:
  (( A\andB )\impC )\imp( A\imp( B\impC )) in F
proof
  set qp = ( B\imp( A\andB ));
  set pr = (( A\andB )\impC)\imp( B\impC );
A1: ( A\imp( qp\imppr ))\imp( ( A\impqp )\imp( A\imppr )) in F by Def38;
A2: A\imp( B\imp( A\andB )) in F by Def38;
  A\imp(( B\imp( A\andB ))\imp((( A\andB )\impC )\imp( B\impC ))) in
  F by Th44,Th39;
  then ( ( A\impqp )\imp( A\imppr )) in F by A1,Def38;
  then
A3: A\imp((( A\andB )\impC )\imp( B\impC )) in F by A2,Def38;
  (A\imp((( A\andB )\impC )\imp( B\impC )))\imp((( A\andB )\impC )\imp(
  A\imp( B\impC ))) in F by Th41;
  hence thesis by A3,Def38;
end;
