reserve IPS for IncProjSp,
  z for POINT of IPS;
reserve IPP for Desarguesian 2-dimensional IncProjSp,
  a,b,c,d,p,pp9,q,o,o9,o99 ,oo9 for POINT of IPP,
  r,s,x,y,o1,o2 for POINT of IPP,
  O1,O2,O3,O4,A,B,C,O,Q,Q1 ,Q2,Q3,R,S,X for LINE of IPP;

theorem Th10:
  not o on A implies IncProj(A,o,A) = id CHAIN(A)
proof
  set f = IncProj(A,o,A);
  assume
A1: not o on A;
A2: for x being object st x in CHAIN(A) holds f.x=x
  proof
    let x be object;
    assume x in CHAIN(A);
    then ex x9 being Element of the Points of IPP st x = x9 & x9 on A;
    hence thesis by A1,PROJRED1:24;
  end;
  dom f = CHAIN(A) by A1,Th4;
  hence thesis by A2,FUNCT_1:17;
end;
