reserve SAS for Semi_Affine_Space;
reserve a,a9,a1,a2,a3,a4,b,b9,c,c9,d,d9,d1,d2,o,p,p1,p2,q,r,r1,r2,s,x, y,t,z
  for Element of SAS;

theorem
  o<>a & o<>b & o,a,b are_collinear & o,a,a9 are_collinear & o,b,b9
  are_collinear implies a,b // a9,b9
proof
  assume that
A1: o<>a and
A2: o<>b and
A3: o,a,b are_collinear and
A4: o,a,a9 are_collinear and
A5: o,b, b9 are_collinear;
A6: now
A7: o,a // o,b by A3;
    o,a // o,a9 by A4;
    then
A8: o,b // o,a9 by A1,A7,Def1;
    o,b // o,b9 by A5;
    then o,a9 // o,b9 by A2,A8,Def1;
    then
A9: o,a9 // a9,b9 by Th7;
    o,b // a,b by A7,Th7;
    then
A10: a,b // o,a9 by A2,A8,Def1;
    assume o<>a9;
    hence thesis by A10,A9,Th8;
  end;
  now
    assume
A11: o=a9;
    then a9,a // a9,b by A3;
    then
A12: a,b // a9,b by Th7;
    a9,b // a9,b9 by A5,A11;
    hence thesis by A2,A11,A12,Th8;
  end;
  hence thesis by A6;
end;
