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 Th4:
  not p,a,b are_collinear & p,b,c are_collinear & p,a,d1 are_collinear &
    p,a,d2 are_collinear & a,b '||' c,d1 & a,b '||' c,d2 implies d1=d2
proof
  assume that
A1: not p,a,b are_collinear and
A2: p,b,c are_collinear and
A3: p,a,d1 are_collinear and
A4: p,a,d2 are_collinear and
A5: a,b '||' c,d1 and
A6: a,b '||' c,d2;
A7: p<>a by A1,DIRAF:31;
A8: a<>b by A1,DIRAF:31;
A9: now
    p,a,a are_collinear by DIRAF:31;
    then
A10: d1,d2,a are_collinear by A3,A4,A7,DIRAF:32;
A11: d1,d2,d1 are_collinear by DIRAF:31;
A12: p,c,b are_collinear by A2,DIRAF:30;
A13: d1,d2,d2 are_collinear by DIRAF:31;
A14: p,a,p are_collinear by DIRAF:31;
    then
A15: d1,d2,p are_collinear by A3,A4,A7,DIRAF:32;
    c,d1 '||' c,d2 by A5,A6,A8,DIRAF:23;
    then
A16: c,d1,d2 are_collinear by DIRAF:def 5;
    then
A17: d1,d2,c are_collinear by DIRAF:30;
    assume
A18: p<>c;
    assume
A19: d1<>d2;
    d1,d2,p are_collinear by A3,A4,A7,A14,DIRAF:32;
    then
A20: p,c,d1 are_collinear by A19,A17,A11,DIRAF:32;
    d1,d2,c are_collinear by A16,DIRAF:30;
    then p,c,d2 are_collinear by A19,A15,A13,DIRAF:32;
    then d1,d2,b are_collinear by A18,A20,A12,DIRAF:32;
    hence contradiction by A1,A19,A15,A10,DIRAF:32;
  end;
A21: p,d2,a are_collinear by A4,DIRAF:30;
A22: p,d1,a are_collinear by A3,DIRAF:30;
  now
A23: p,d2,p are_collinear by DIRAF:31;
    assume
A24: c =p;
    then
A25: p,d2 '||' a,b by A6,DIRAF:22;
A26: now
      assume
A27:  p<>d2;
      then p,d2,b are_collinear by A21,A25,DIRAF:33;
      hence contradiction by A1,A21,A23,A27,DIRAF:32;
    end;
A28: p,d1,p are_collinear by DIRAF:31;
A29: p,d1 '||' a,b by A5,A24,DIRAF:22;
    now
      assume
A30:  p<>d1;
      then p,d1,b are_collinear by A22,A29,DIRAF:33;
      hence contradiction by A1,A22,A28,A30,DIRAF:32;
    end;
    hence thesis by A26;
  end;
  hence thesis by A9;
end;
