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
  for A be being_line Subset of AS st
  a in A & b in A & c in A & a<>b & a,b // c,d holds d in A
proof
  let A be being_line Subset of AS;
  assume that
A1: a in A and
A2: b in A and
A3: c in A and
A4: a<>b and
A5: a,b // c,d;
  now
    set C=Line(c,d);
A6: c in C by Th14;
A7: d in C by Th14;
    assume
A8: c <>d;
    then C is being_line;
    then A // C by A1,A2,A4,A5,A8,A6,A7,Th37;
    hence thesis by A3,A6,A7,Th44;
  end;
  hence thesis by A3;
end;
