
theorem Th2:
  for OAS being OAffinSpace holds for a,b,c being (Element of OAS),
a9,b9,c9 being (Element of Lambda(OAS)) st a=a9 & b=b9 & c =c9
 holds a,b,c are_collinear
  iff LIN a9,b9,c9
proof
  let OAS be OAffinSpace;
  let a,b,c be (Element of OAS), a9,b9,c9 be (Element of Lambda(OAS)) such
  that
A1: a=a9 & b=b9 & c =c9;
A2: now
    assume a,b,c are_collinear;
    then a,b '||' a,c by DIRAF:def 5;
    then a9,b9 // a9, c9 by A1,DIRAF:38;
    hence LIN a9,b9,c9 by AFF_1:def 1;
  end;
  now
    assume LIN a9,b9,c9;
    then a9,b9 // a9,c9 by AFF_1:def 1;
    then a,b '||' a,c by A1,DIRAF:38;
    hence a,b,c are_collinear by DIRAF:def 5;
  end;
  hence thesis by A2;
end;
