reserve OAS for OAffinSpace;
reserve a,a9,b,b9,c,c9,d,d1,d2,e1,e2,e3,e4,e5,e6,p,p9,q,r,x,y,z for Element of
  OAS;

theorem
  not p,a,b are_collinear & a,p // p,a9 & b,p // p,b9 & a,b '||' a9,b9
  implies a,b
  // b9,a9
proof
  assume that
A1: not p,a,b are_collinear and
A2: a,p // p,a9 and
A3: b,p // p,b9 and
A4: a,b '||' a9,b9;
A5: not p,b,a are_collinear by A1,DIRAF:30;
  Mid b,p,b9 by A3,DIRAF:def 3;
  then b,p,b9 are_collinear by DIRAF:28;
  then
A6: p,b,b9 are_collinear by DIRAF:30;
  Mid a,p,a9 by A2,DIRAF:def 3;
  then a,p,a9 are_collinear by DIRAF:28;
  then
A7: p,a,a9 are_collinear by DIRAF:30;
A8: b,a '||' a9,b9 by A4,DIRAF:22;
  a<>p by A1,DIRAF:31;
  then consider q such that
A9: b,p // p,q and
A10: b,a // a9,q by A2,ANALOAF:def 5;
  Mid b,p,q by A9,DIRAF:def 3;
  then b,p,q are_collinear by DIRAF:28;
  then
A11: p,b,q are_collinear by DIRAF:30;
  b,a '||' a9,q by A10,DIRAF:def 4;
  then b,a // a9,b9 by A10,A5,A7,A6,A11,A8,Th4;
  hence thesis by DIRAF:2;
end;
