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 Th15:
  c in Line(a,b) & d in Line(a,b) & c <>d implies Line(c,d) c= Line(a,b)
proof
  assume that
A1: c in Line(a,b) and
A2: d in Line(a,b) and
A3: c <>d;
A4: LIN a,b,d by A2,Def2;
A5: LIN a,b,c by A1,Def2;
    let x be object;
    assume
A6: x in Line(c,d);
    then reconsider x9=x as Element of AS;
    LIN c,d,x9 by A6,Def2;
    then LIN a,b,x9 by A3,A5,A4,Th10;
    hence x in Line(a,b) by Def2;
end;
