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 Th5:
  not a,b,c are_collinear & a,b '||' c,d1 & a,b '||' c,d2 & a,c '||' b,d1 &
  a,c '||' b,d2 implies d1=d2
proof
  assume that
A1: not a,b,c are_collinear and
A2: a,b '||' c,d1 and
A3: a,b '||' c,d2 and
A4: a,c '||' b,d1 and
A5: a,c '||' b,d2;
  assume
A6: d1<>d2;
  a<>c by A1,DIRAF:31;
  then b,d1 '||' b,d2 by A4,A5,DIRAF:23;
  then b,d1,d2 are_collinear by DIRAF:def 5;
  then
A7: d1,d2,b are_collinear by DIRAF:30;
A8: now
    assume c =d1;
    then c,a '||' c,b by A4,DIRAF:22;
    then c,a,b are_collinear by DIRAF:def 5;
    hence contradiction by A1,DIRAF:30;
  end;
A9: d1,d2,d1 are_collinear by DIRAF:31;
  a<>b by A1,DIRAF:31;
  then c,d1 '||' c,d2 by A2,A3,DIRAF:23;
  then
A10: c,d1,d2 are_collinear by DIRAF:def 5;
  then
A11: d1,d2,c are_collinear by DIRAF:30;
  d1,d2,c are_collinear by A10,DIRAF:30;
  then d1,d2 '||' c,d1 by A9,DIRAF:34;
  then d1,d2 '||' a,b or c =d1 by A2,DIRAF:23;
  then d1,d2 '||' b,a by A8,DIRAF:22;
  then d1,d2,a are_collinear by A6,A7,DIRAF:33;
  hence contradiction by A1,A6,A11,A7,DIRAF:32;
end;
