reserve F for Field;
reserve S for OrtSp of F;
reserve a,b,c,d,p,q,r,x,y,z for Element of S;
reserve k,l for Element of F;

theorem
  not b _|_ a implies PProJ(a,b,x,y) = PProJ(a,b,y,x)
proof
  assume
A1: not b _|_ a;
A2: now
    assume not x _|_ y;
    then
A3: x <> 0.S & y <> 0.S by Th1,Th2;
    a <> 0.S by A1,Th1,Th2;
    then
    ex r st not r _|_ a & not r _|_ x & not r _|_ y & not r _|_ a by A3,Def1;
    then consider r such that
A4: not r _|_ a and
A5: not r _|_ x and
A6: not r _|_ y;
A7: not y _|_ r by A6,Th2;
    PProJ(a,b,y,x) = ProJ(a,b,r)*ProJ(r,a,y)*ProJ(y,r,x) by A1,A4,A6,Def3;
    then
A8: PProJ(a,b,y,x) = ProJ(a,b,r)*(ProJ(r,a,y)*ProJ (y,r,x)) by GROUP_1:def 3;
    ( not a _|_ r)& not x _|_ r by A4,A5,Th2;
    then
A9: PProJ(a,b,y,x) = ProJ(a,b,r)*(ProJ(r,a,x)*ProJ(x,r,y)) by A7,A8,Th27;
    PProJ(a,b,x,y) = ProJ(a,b,r)*ProJ(r,a,x)*ProJ(x,r,y) by A1,A4,A5,Def3;
    hence thesis by A9,GROUP_1:def 3;
  end;
  now
    assume x _|_ y;
    then y _|_ x & PProJ(a,b,y,x) = 0.F by A1,Th2,Th29;
    hence thesis by A1,Th29;
  end;
  hence thesis by A2;
end;
