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 Th6:
  not p,b,c are_collinear & Mid b,p,a & p,c,d are_collinear & b,c '||' d,a
    implies Mid c,p,d
proof
  assume that
A1: not p,b,c are_collinear and
A2: Mid b,p,a and
A3: p,c,d are_collinear and
A4: b,c '||' d,a;
A5: p,d,c are_collinear by A3,DIRAF:30;
  b,p,a are_collinear by A2,DIRAF:28;
  then
A6: p,b,a are_collinear by DIRAF:30;
  p,c '||' p,d by A3,DIRAF:def 5;
  then
A7: p,c // p,d or p,c // d,p by DIRAF:def 4;
  assume
A8: not Mid c,p,d;
  then
A9: d<>p by DIRAF:10;
A10: now
    assume p,c // d,p;
    then c,p // p,d by DIRAF:2;
    hence contradiction by A8,DIRAF:def 3;
  end;
  p<>c by A8,DIRAF:10;
  then consider q such that
A11: p,b // p,q and
A12: b,c '||' d,q and
  d<>q by A9,A7,A10,Th2;
A13: p,d,p are_collinear by DIRAF:31;
  p,b '||' p,q by A11,DIRAF:def 4;
  then p,b,q are_collinear by DIRAF:def 5;
  then a=q by A1,A3,A4,A12,A6,Th4;
  then b,p // p,q by A2,DIRAF:def 3;
  then p,q // b,p by DIRAF:2;
  then p,b // b,p or p=q by A11,DIRAF:3;
  then p=b or p=q by ANALOAF:def 5;
  then p,d '||' c,b by A1,A12,DIRAF:22,31;
  then p,d,b are_collinear by A9,A5,DIRAF:33;
  hence contradiction by A1,A9,A5,A13,DIRAF:32;
end;
