theorem Th4:
  A is being_line & a in A & b in A & c in A implies LIN a,b,c
proof
  assume that
A1: A is being_line and
A2: a in A and
A3: b in A and
A4: c in A;
  reconsider a9=a,b9=b,c9=c as Element of the AffinStruct of X;
  reconsider A9=A as Subset of the AffinStruct of X;
  A9 is being_line by A1,ANALMETR:43;
  then LIN a9,b9,c9 by A2,A3,A4,AFF_1:21;
  hence thesis by ANALMETR:40;
end;
