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
  not LIN o,a,b & LIN o,a,a9 & LIN o,b,b9 & LIN o,b,x & a,b // a9,b9 & a
  ,b // a9,x implies b9=x
proof
  assume that
A1: not LIN o,a,b and
A2: LIN o,a,a9 and
A3: LIN o,b,b9 and
A4: LIN o,b,x and
A5: a,b // a9,b9 and
A6: a,b // a9,x;
  set A=Line(o,a), C=Line(o,b), P=Line(a9,b9);
A7: a9 in P by Th14;
  assume
A8: b9<>x;
A9: a9<>b9
  proof
    assume
A10: a9=b9;
    then a9=o by A1,A2,A3,Th53;
    hence contradiction by A1,A4,A6,A8,A10,Th54;
  end;
  then
A11: P is being_line;
A12: o<>b by A1,Th6;
  then
A13: C is being_line;
A14: b9 in P by Th14;
  a<>b by A1,Th6;
  then a9,b9 // a9,x by A5,A6,Th4;
  then LIN a9,b9,x;
  then
A15: x in P by A9,A11,A7,A14,Th24;
A16: b in C by Th14;
A17: o in C by Th14;
  then
A18: x in C by A4,A12,A13,A16,Th24;
  b9 in C by A3,A12,A13,A17,A16,Th24;
  then
A19: a9 in C by A8,A13,A11,A7,A14,A18,A15,Th17;
A20: o<>a by A1,Th6;
  then
A21: A is being_line;
A22: a9<>o
  proof
    assume
A23: a9=o;
    then b9=o by A1,A3,A5,Th54;
    hence contradiction by A1,A4,A6,A8,A23,Th54;
  end;
A24: o in A by Th14;
A25: a in A by Th14;
  then a9 in A by A2,A20,A21,A24,Th24;
  then b in A by A22,A21,A13,A24,A17,A16,A19,Th17;
  hence contradiction by A1,A21,A24,A25,Th20;
end;
