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 Th8:
  not o on A & not o on B implies IncProj(A,o,B)" = IncProj(B,o,A)
proof
  set f=IncProj(A,o,B), g=IncProj (B,o,A);
  assume
A1: ( not o on A)& not o on B;
  then
A2: rng f = CHAIN(B) by Th5;
A3: dom f = CHAIN(A) by A1,Th4;
A4: now
    let y,x be object;
A5: now
      assume that
A6:   x in dom f and
A7:   y=f.x;
      consider x9 being POINT of IPP such that
A8:   x = x9 & x9 on A by A3,A6;
      reconsider y9=y as POINT of IPP by A1,A7,A8,PROJRED1:19;
A9:   y9 on B by A1,A7,A8,PROJRED1:20;
      then
A10:  y in CHAIN(B);
      ex O st o on O & x9 on O & y9 on O by A1,A7,A8,A9,PROJRED1:def 1;
      hence y in rng f & x=g.y by A1,A8,A9,A10,Th5,PROJRED1:def 1;
    end;
    now
      assume that
A11:  y in rng f and
A12:  x=g.y;
      consider y9 being POINT of IPP such that
A13:  y = y9 & y9 on B by A2,A11;
      reconsider x9=x as POINT of IPP by A1,A12,A13,PROJRED1:19;
A14:  x9 on A by A1,A12,A13,PROJRED1:20;
      then ex O st o on O & y9 on O & x9 on O by A1,A12,A13,PROJRED1:def 1;
      hence x in dom f & y=f.x by A1,A13,A14,PROJRED1:def 1;
    end;
    hence y in rng f & x=g.y iff x in dom f & y=f.x by A5;
  end;
A15: f is one-to-one by A1,Th7;
  dom g = CHAIN(B) by A1,Th4
    .= rng f by A1,Th5;
  hence thesis by A15,A4,FUNCT_1:32;
end;
