theorem
  p <==> q & q <==> r implies p <==> r
proof
  assume that
A1: p <==> q and
A2: q <==> r;
A3: r => q is valid by A2,Th50;
  q => p is valid by A1,Th50;
  then
A4: r => p is valid by A3,LUKASI_1:42;
A5: q => r is valid by A2,Th50;
  p => q is valid by A1,Th50;
  then p => r is valid by A5,LUKASI_1:42;
  hence thesis by A4,Th50;
end;
