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