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 Th91:
  A\iffB in F & B\iffC in F implies A\iffC in F
  proof
    assume
A1: A\iffB in F & B\iffC in F;
    A\iffB\imp(A\impB) in F & A\iffB\imp(B\impA) in F &
    B\iffC\imp(B\impC) in F & B\iffC\imp(C\impB) in F by Def38; then
    A\impB in F & B\impA in F & C\impB in F & B\impC in F by A1,Def38; then
    A\impC in F & C\impA in F by Th45; then
A2: (A\impC)\and(C\impA) in F by Th35;
    (A\impC)\and(C\impA)\imp(A\iffC) in F by Def38;
    hence thesis by A2,Def38;
  end;
