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 Th7:
  not o on A & not o on B implies IncProj(A,o,B) is one-to-one
proof
  set f = IncProj(A,o,B);
  assume
A1: ( not o on A)& not o on B;
  now
    let x1,x2 be object;
    assume that
A2: x1 in dom f and
A3: x2 in dom f and
A4: f.x1 = f.x2;
    x1 in CHAIN(A) by A1,A2,Th4;
    then consider a such that
A5: x1=a and
A6: a on A;
    x2 in CHAIN(A) by A1,A3,Th4;
    then consider b such that
A7: x2=b and
A8: b on A;
    reconsider x = f.a,y = f.b as POINT of IPP by A1,A6,A8,PROJRED1:19;
    x=y by A4,A5,A7;
    hence x1 = x2 by A1,A5,A6,A7,A8,PROJRED1:23;
  end;
  hence thesis by FUNCT_1:def 4;
end;
