
theorem Th8:
  for OAS being OAffinSpace holds for o,a,b,a9,b9 being Element
 of OAS st not o,a,b are_collinear & o,a // o,a9 & o,b,b9 are_collinear &
a,b '||' a9,b9
  holds o,b // o,b9 & a,b // a9,b9
proof
  let OAS be OAffinSpace;
  let o,a,b,a9,b9 be Element of OAS such that
A1: not o,a,b are_collinear and
A2: o,a // o,a9 and
A3: o,b,b9 are_collinear and
A4: a,b '||' a9,b9;
A5: o<>a by A1,DIRAF:31;
  consider a2 being Element of OAS such that
A6: Mid a,o,a2 and
A7: o<>a2 by DIRAF:13;
  a,o // o,a2 by A6,DIRAF:def 3;
  then consider b2 being Element of OAS such that
A8: b,o // o,b2 and
A9: b,a // a2,b2 by A5,ANALOAF:def 5;
A10: o,b // b2,o by A8,DIRAF:2;
  a,o // o,a2 by A6,DIRAF:def 3;
  then a2,o // o,a by DIRAF:2;
  then
A11: a2,o // o,a9 by A2,A5,DIRAF:3;
  a,o,a2 are_collinear by A6,DIRAF:28;
  then
A12: o,a2,a are_collinear by DIRAF:30;
A13: o<>b2
  proof
    assume o=b2;
    then o,a2 // a,b by A9,DIRAF:2;
    then o,a2 '||' a,b by DIRAF:def 4;
    then o,a2,o are_collinear & o,a2,b are_collinear by A7,A12,DIRAF:31,33;
    hence contradiction by A1,A7,A12,DIRAF:32;
  end;
  Mid b,o,b2 by A8,DIRAF:def 3;
  then b,o,b2 are_collinear by DIRAF:28;
  then
A14: b2,o,b are_collinear by DIRAF:30;
A15: not o,a2,b2 are_collinear
  proof
A16: b2,o,o are_collinear by DIRAF:31;
A17: o,a2,o are_collinear by DIRAF:31;
    assume o,a2,b2 are_collinear;
    then b2,o,a are_collinear by A7,A12,A17,DIRAF:32;
    hence contradiction by A1,A14,A13,A16,DIRAF:32;
  end;
  a2,b2 // b,a by A9,DIRAF:2;
  then
A18: a2,b2 '||' a,b by DIRAF:def 4;
  b<>a by A1,DIRAF:31;
  then
A19: a2,b2 '||' a9,b9 by A4,A18,DIRAF:23;
A20: a,b // b2,a2 by A9,DIRAF:2;
  Mid b,o,b2 by A8,DIRAF:def 3;
  then b,o,b2 are_collinear by DIRAF:28;
  then
A21: o,b,b2 are_collinear by DIRAF:30;
A22: o,b,o are_collinear by DIRAF:31;
  o<>b by A1,DIRAF:31;
  then
A23: o,b2,b9 are_collinear by A3,A21,A22,DIRAF:32;
  then a2,b2 // b9,a9 by A15,A11,A19,Th7;
  then
A24: b2,a2 // a9,b9 by DIRAF:2;
  b2,o // o,b9 by A15,A11,A19,A23,Th7;
  hence o,b // o,b9 by A13,A10,DIRAF:3;
  a2<>b2 by A15,DIRAF:31;
  hence thesis by A20,A24,DIRAF:3;
end;
