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
  a,b '||' c,d & not a,b,c are_collinear & p,a,d are_collinear &
   p,b,c are_collinear implies not p,a,b are_collinear
proof
  assume that
A1: a,b '||' c,d and
A2: not a,b,c are_collinear and
A3: p,a,d are_collinear and
A4: p,b,c are_collinear;
A5: p,a,a are_collinear by DIRAF:31;
  assume p,a,b are_collinear;
  then
A6: a,b,d are_collinear by A2,A3,A4,A5,DIRAF:32;
A7: a,b '||' d,c by A1,DIRAF:22;
  a<>b by A2,DIRAF:31;
  hence contradiction by A2,A6,A7,DIRAF:33;
end;
