reserve CPS for proper CollSp,
  B for Subset of CPS,
  p for Point of CPS,
  x, y, z, Y for set;
reserve a,b,c,p,q for POINT of IncProjSp_of(CPS),
  P,Q for LINE of IncProjSp_of(CPS),
  a9,b9,c9,p9,q9,r9 for Point of CPS,
  P9 for LINE of CPS;
reserve CPS for CollProjectiveSpace,
  a,b,c,d,p,q for POINT of IncProjSp_of(CPS ),
  P,Q,S,M,N for LINE of IncProjSp_of(CPS),
  a9,b9,c9,d9,p9,q9 for Point of CPS;

theorem Th12:
  ex a, b, c st a<>b & b<>c & c <>a & a on P & b on P & c on P
proof
  reconsider P9= P as LINE of CPS by Th1;
  consider a99, b99 being Point of CPS such that
A1: a99<>b99 and
A2: P9 = Line(a99,b99) by COLLSP:def 7;
  consider c9 such that
A3: a99<>c9 & b99<>c9 and
A4: a99, b99, c9 are_collinear by ANPROJ_2:def 10;
  reconsider a=a99, b=b99, c =c9 as POINT of IncProjSp_of(CPS);
  take a,b,c;
  thus a<>b & b<>c & c <>a by A1,A3;
  a99 in P9 by A2,COLLSP:10;
  then
A5: a on P by Th5;
  b99 in P9 by A2,COLLSP:10;
  then
A6: b on P by Th5;
  ex Q st a on Q & b on Q & c on Q by A4,Th10;
  hence thesis by A1,A5,A6,Th8;
end;
