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
  p,b '||' p,c & p<>b implies ex d st p,a '||' p,d & a,b '||' c,d
proof
  assume that
A1: p,b '||' p,c and
A2: p<>b;
A3: now
    assume
A4: p<>c;
A5: now
      assume p,b // c,p;
      then b,p // p,c by DIRAF:2;
      then consider d such that
A6:   a,p // p,d and
A7:   a,b '||' c,d and
      c <>d and
      d<>p by A2,A4,Th1;
      p,a // d,p by A6,DIRAF:2;
      then p,a '||' p,d by DIRAF:def 4;
      hence thesis by A7;
    end;
    now
      assume p,b // p,c;
      then consider d such that
A8:   p,a // p,d and
A9:   a,b '||' c,d and
      c <>d by A2,A4,Th2;
      p,a '||' p,d by A8,DIRAF:def 4;
      hence thesis by A9;
    end;
    hence thesis by A1,A5,DIRAF:def 4;
  end;
  now
A10: a,b '||' p,p by DIRAF:20;
A11: p,a '||' p,p by DIRAF:20;
    assume p=c;
    hence thesis by A11,A10;
  end;
  hence thesis by A3;
end;
