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,l*y) = l*PProJ(a,b,x,y)
proof
  set 0F = 0.F;
  assume
A1: not b _|_ a;
A2: now
    assume not x _|_ y;
    then
A3: x <> 0.S by Th1;
    a <> 0.S by A1,Th1,Th2;
    then ex p st not p _|_ a & not p _|_ x & not p _|_ a & not p _|_ x by A3
,Def1;
    then consider p such that
A4: not p _|_ a and
A5: not p _|_ x;
    PProJ(a,b,x,l*y) = ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,l*y) by A1,A4,A5,Def3;
    then
A6: PProJ(a,b,x,l*y) = (l*ProJ(x,p,y))*(ProJ(a,b,p)*ProJ(p,a,x)) by A5,Th12;
    PProJ(a,b,x,y) = ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,y) by A1,A4,A5,Def3;
    hence thesis by A6,GROUP_1:def 3;
  end;
  now
    assume
A7: x _|_ y;
    then y _|_ x by Th2;
    then l*y _|_ x by Def1;
    then
A8: PProJ(a,b,x,l*y) = 0F by A1,Th29;
    y _|_ x by A7,Th2;
    then l*PProJ(a,b,x,y) = l*0F by A1,Th29;
    hence thesis by A8;
  end;
  hence thesis by A2;
end;
