theorem Th2:
  p,b // p,c & p<>c & b<>p implies ex d st p,a // p,d & a,b '||' c, d & c <>d
proof
  assume that
A1: p,b // p,c and
A2: p<>c and
A3: b<>p;
  consider q such that
A4: Mid b,p,q and
A5: p<>q by DIRAF:13;
A6: b,p // p,q by A4,DIRAF:def 3;
  then consider r such that
A7: a,p // p,r and
A8: a,b '||' q,r and
A9: q<>r and
A10: r<>p by A3,A5,Th1;
  b,p // c,p by A1,DIRAF:2;
  then p,q // c,p by A3,A6,ANALOAF:def 5;
  then q,p // p,c by DIRAF:2;
  then consider d such that
A11: r,p // p,d and
A12: r,q '||' c,d and
A13: c <>d and
  d<>p by A2,A5,Th1;
  p,r // d,p by A11,DIRAF:2;
  then a,p // d,p by A7,A10,DIRAF:3;
  then
A14: p,a // p,d by DIRAF:2;
  q,r '||' c,d by A12,DIRAF:22;
  then a,b '||' c,d by A8,A9,DIRAF:23;
  hence thesis by A13,A14;
end;
