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;

theorem Th5:
  p = p9 & P = P9 implies (p on P iff p9 in P9)
proof
  reconsider P99= P9 as LINE of IncProjSp_of(CPS) by Th1;
  reconsider P999= P99 as Element of ProjectiveLines(CPS);
  assume
A1: p = p9 & P = P9;
  hereby
    assume p on P;
    then [p9,P9] in Proj_Inc(CPS) by A1;
    then ex Y st P9= Y & p9 in Y by Def2;
    hence p9 in P9;
  end;
  assume p9 in P9;
  then [p9,P999] in Proj_Inc(CPS) by Def2;
  hence thesis by A1;
end;
