reserve SAS for AffinPlane;

theorem
  for o,a being Element of SAS holds ex p being Element of SAS st for b,
  c being Element of SAS holds (o,a // o,p & ex d being Element of SAS st ( o,p
  // o,b implies o,c // o,d & p,c // b,d ))
proof
  let o,a be Element of SAS;
  ex p being Element of SAS st o<>p & o,a // o,p
  proof
    consider x,y being Element of SAS such that
A1: x<>y by DIRAF:40;
    now
      assume a=o;
      then
A2:   o,a // o,x & o,a // o,y by AFF_1:3;
      o<>x or o<>y by A1;
      hence thesis by A2;
    end;
    hence thesis by AFF_1:2;
  end;
  then consider p being Element of SAS such that
A3: o<>p and
A4: o,a // o,p;
  take p;
  thus for b,c being Element of SAS holds (o,a // o,p & ex d being Element of
  SAS st ( o,p // o,b implies o,c // o,d & p,c // b,d ))
  proof
    let b,c be Element of SAS;
    ex d being Element of SAS st (o,p // o,b implies o,c // o,d & p,c // b,d)
    proof
      now
        assume o,p // o,b;
        then p,o // o,b by AFF_1:4;
        then consider d being Element of SAS such that
A5:     c,o // o,d & c,p // b,d by A3,DIRAF:40;
        o,c // o,d & p,c // b,d by A5,AFF_1:4;
        hence thesis;
      end;
      hence thesis;
    end;
    hence thesis by A4;
  end;
end;
