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 Th29:
  for A be being_line Subset of AS holds
  (a,b // A iff ex c,d st c <>d & c in A & d in A & a,b // c,d )
proof
A1: a,b // A implies ex c,d st c <>d & c in A & d in A & a,b // c,d by Th27;
  let A be being_line Subset of AS;
  (ex c,d st c <>d & c in A & d in A & a,b // c,d) implies a,b // A
  proof
    assume ex c,d st c <>d & c in A & d in A & a,b // c,d;
    then consider c,d such that
A2: c <>d and
A3: c in A and
A4: d in A and
A5: a,b // c,d;
    A=Line(c,d) by A2,A3,A4,Lm6;
    hence thesis by A2,A5;
  end;
  hence thesis by A1;
end;
