theorem B124and234Ordered:
  between a,b,d & between b,c,d implies a,b,c,d are_ordered
   proof
     assume that
H1:  between a,b,d and
H2:  between b,c,d;
X1:  between a,b,c by H1, H2, B124and234then123;
     per cases;
     suppose b = c;
       hence thesis by H1, Bqaa, Baaq;
     end;
     suppose b <> c;
       hence thesis by X1, H2, BTransitivityOrdered;
     end;
   end;
