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 Th57:
  A\impB\imp(\notB\imp\notA) in F
  proof
    A\impB\imp(A\imp\notB\imp\notA) in F by Def38; then
A1: A\imp\notB\imp(A\impB\imp\notA) in F by Th38;
    \notB\imp(A\imp\notB) in F by Def38; then
    \notB\imp(A\impB\imp\notA) in F by A1,Th45;
    hence thesis by Th38;
  end;
