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,a9 are_collinear & p,a // p,b & p,a9 // p,b9 & a,a9 '||' b,b9
  implies a,a9 // b,b9
proof
  assume that
A1: not p,a,a9 are_collinear and
A2: p,a // p,b and
A3: p,a9 // p,b9 and
A4: a,a9 '||' b,b9;
  consider c such that
A5: Mid a,p,c and
A6: p<>c by DIRAF:13;
A7: a,p // p,c by A5,DIRAF:def 3;
A8: a<>p by A1,DIRAF:31;
  then consider c9 such that
A9: a9,p // p,c9 and
A10: a9,a // c,c9 by A7,ANALOAF:def 5;
A11: a9,a '||' c9,c by A10,DIRAF:def 4;
A12: c <>c9
  proof
    assume c =c9;
    then Mid a9,p,c by A9,DIRAF:def 3;
    then a9,p,c are_collinear by DIRAF:28;
    then
A13: p,c,a9 are_collinear by DIRAF:30;
    a,p,c are_collinear by A5,DIRAF:28;
    then
A14: p,c,a are_collinear by DIRAF:30;
    p,c,p are_collinear by DIRAF:31;
    hence contradiction by A1,A6,A14,A13,DIRAF:32;
  end;
  p,a // c,p by A7,DIRAF:2;
  then c,p // p,b by A2,A8,ANALOAF:def 5;
  then consider b99 be Element of OAS such that
A15: c9,p // p,b99 and
A16: c9,c // b,b99 by A6,ANALOAF:def 5;
A17: a9,a '||' b,b9 by A4,DIRAF:22;
A18: p<>c9
  proof
    assume p=c9;
    then a9,a '||' c,p by A10,DIRAF:def 4;
    then
A19: p,c '||' a,a9 by DIRAF:22;
    a,p '||' p,c by A7,DIRAF:def 4;
    then a,p '||' a,a9 by A6,A19,DIRAF:23;
    then a,p,a9 are_collinear by DIRAF:def 5;
    hence contradiction by A1,DIRAF:30;
  end;
  p,a '||' p,b by A2,DIRAF:def 4;
  then
A20: p,a,b are_collinear by DIRAF:def 5;
A21: c9,c // a,a9 by A10,DIRAF:2;
  a9,p '||' p,c9 by A9,DIRAF:def 4;
  then
A22: p,a9 '||' p,c9 by DIRAF:22;
  p,a9 '||' p,b9 by A3,DIRAF:def 4;
  then
A23: p,a9,b9 are_collinear by DIRAF:def 5;
  c9,p '||' p,b99 by A15,DIRAF:def 4;
  then p,c9 '||' p,b99 by DIRAF:22;
  then p,a9 '||' p,b99 by A18,A22,DIRAF:23;
  then
A24: p,a9,b99 are_collinear by DIRAF:def 5;
  c9,c '||' b,b99 by A16,DIRAF:def 4;
  then
A25: a9,a '||' b,b99 by A12,A11,DIRAF:23;
  not p,a9,a are_collinear by A1,DIRAF:30;
  then c9,c // b,b9 by A20,A23,A17,A16,A24,A25,Th4;
  hence thesis by A12,A21,ANALOAF:def 5;
end;
