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 Th4:
  not o on A & not o on B implies dom IncProj(A,o,B) = CHAIN(A)
proof
  assume
A1: ( not o on A)& not o on B;
A2: now
    let x be object;
    assume
A3: x in dom IncProj(A,o,B);
    then reconsider x9 = x as POINT of IPP;
    x9 on A by A1,A3,PROJRED1:def 1;
    hence x in CHAIN(A);
  end;
  now
    let x be object;
    assume x in CHAIN(A);
    then ex b st b=x & b on A;
    hence x in dom IncProj(A,o,B) by A1,PROJRED1:def 1;
  end;
  hence thesis by A2,TARSKI:2;
end;
