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 Th53:
  not LIN o,a,b & LIN o,a,a9 & LIN o,b,b9 & a9=b9 implies a9=o & b9=o
proof
  assume that
A1: not LIN o,a,b and
A2: LIN o,a,a9 and
A3: LIN o,b,b9 and
A4: a9=b9;
  set A=Line(o,a), C=Line(o,b);
A5: o in A by Th14;
A6: o<>b by A1,Th6;
  then
A7: C is being_line;
A8: o<>a by A1,Th6;
  then
A9: A is being_line;
A10: a in A by Th14;
  then
A11: a9 in A by A2,A8,A9,A5,Th24;
A12: b in C by Th14;
A13: o in C by Th14;
  then
A14: b9 in C by A3,A6,A7,A12,Th24;
  A<>C by A1,A9,A5,A10,A12,Th20;
  hence thesis by A4,A9,A7,A5,A13,A14,A11,Th17;
end;
