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 Th45:
  A\impB in F & B\impC in F implies A\impC in F
  proof
    assume that
A1: A\impB in F and
A2: B\impC in F;
    (A\impB)\imp((B\impC)\imp(A\impC)) in F by Th39;
  then (B\impC)\imp(A\impC) in F by A1,Def38;
  hence thesis by A2,Def38;
end;
