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