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
  a9,b9 // K & LIN p,a,a9 & LIN p,b,b9 & p in K & not a in K & a=b
  implies a9=b9
proof
  assume that
A1: a9,b9 // K and
A2: LIN p,a,a9 and
A3: LIN p,b,b9 and
A4: p in K and
A5: not a in K and
A6: a=b;
  set A=Line(p,a);
A7: b9 in A by A3,A6,Def2;
A8: p in A by Th14;
A9: a9 in A by A2,Def2;
  assume
A10: a9<>b9;
  A is being_line by A4,A5;
  then A=Line(a9,b9) by A9,A7,A10,Lm6;
  then A // K by A1,A10;
  then A=K by A4,A8,Th44;
  hence contradiction by A5,Th14;
end;
