theorem
  between4 a,b,c,d & between b,d,p implies between4 b,c,d,p &
  (b <> d implies between5 a,b,c,d,p)
  proof
    assume
A1: between4 a,b,c,d & between b,d,p;
    hence between4 b,c,d,p by Satz3p6p1,Satz3p6p2;
    thus b <> d implies between5 a,b,c,d,p
    proof
      assume b <> d;
      then between a,b,p & between a,d,p by A1,Satz3p7p1,Satz3p7p2;
      hence thesis by A1,Satz3p6p1,Satz3p6p2;
    end;
  end;
