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 ex C st a in C & A // C
proof
  let A be being_line Subset of AS;
  consider p,q such that
A1: p<>q and
A2: A=Line(p,q) by Def3;
  consider b such that
A3: p,q // a,b and
A4: a<>b by DIRAF:40;
  set C=Line(a,b);
A5: a in C by Th14;
  A // C by A1,A2,A3,A4,Th36;
  hence thesis by A5;
end;
