reserve AS for AffinSpace;
reserve a,a9,b,b9,c,d,o,p,q,r,s,x,y,z,t,u,w for Element of AS;
reserve A,C,D,K for Subset of AS;

theorem Th21:
  c in Line(a,b) & a<>b implies (d in Line(a,b) iff a,b // c,d)
proof
  assume that
A1: c in Line(a,b) and
A2: a<>b;
A3: LIN a,b,c by A1,Def2;
  thus d in Line(a,b) implies a,b // c,d
  proof
    assume d in Line(a,b);
    then LIN a,b,d by Def2;
    hence thesis by A3,Th9;
  end;
  assume a,b // c,d;
  then LIN a,b,d by A2,A3,Th8;
  hence thesis by Def2;
end;
