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 Th48:
  ( A\imp( B\impC ))\imp(( A\andB )\impC ) in F
proof
A1: (( A\andB )\impB)\imp(( B\impC )\imp(( A\andB )\impC )) in F by
Th39;
  ( A\andB )\impB in F by Def38;
  then ( B\impC )\imp(( A\andB )\impC ) in F by A1,Def38;
  then
A2: A\imp(( B\impC )\imp(( A\andB )\impC )) in F by Th44;
A3: ( A\imp(( A\andB )\impC ))\imp((A\andB )\imp( A\impC )) in F by
Th41;
  A\imp(( B\impC )\imp(( A\andB )\impC ))\imp((A\imp( B\impC ))\imp( A\imp
  (( A\andB )\impC ))) in F by Def38;
  then (A\imp( B\impC ))\imp( A\imp(( A\andB )\impC )) in F by A2,Def38;
  then
A4: (A\imp( B\impC ))\imp((A\andB )\imp( A\impC )) in F by A3,Th45;
A5: ( A\andB )\impA in F by Def38;
  ((A\andB )\imp( A\impC ))\imp((( A\andB )\impA )\imp(( A\andB )\impC
  )) in F by Def38;
  then ((A\andB )\imp( A\impC ))\imp(( A\andB )\impC ) in F by A5,Th46;
  hence thesis by A4,Th45;
end;
