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 Th41: :: Contraposition
  (A\imp(B\impC))\imp(B\imp(A\impC)) in F
  proof
A1: B\imp(A\impB) in F by Def38;
    (A\imp(B\impC))\imp((A\impB)\imp(A\impC)) in F by Def38;
    then
A2: (A\impB)\imp((A\imp(B\impC))\imp(A\impC)) in F by Th38;
    ((A\impB)\imp((A\imp(B\impC))\imp(A\impC)))\imp((B\imp(A\impB))\imp(B\imp
    ((A\imp(B\impC))\imp(A\impC)))) in F by Th37;
    then
    (B\imp(A\impB))\imp(B\imp((A\imp(B\impC))\imp(A\impC))) in F by A2,Def38;
    then (B\imp((A\imp(B\impC))\imp(A\impC))) in F by A1,Def38;
    hence thesis by Th38;
  end;
