reserve F for Field;
reserve S for SymSp of F;
reserve a,b,c,d,a9,b9,p,q,r,s,x,y,z for Element of S;
reserve k,l for Element of F;

theorem
  1_F+1_F<>0.F & not b _|_ a implies PProJ(a,b,x,l*y) = l*PProJ(a,b,x,y)
proof
  set 0F = 0.F;
  assume that
A1: 1_F+1_F<>0.F and
A2: not b _|_ a;
A3: now
    assume not x _|_ y;
    then
A4: x <> 0.S by Th1;
    a <> 0.S by A2,Th1,Th2;
    then consider p such that
A5: not p _|_ a and
A6: not p _|_ x by A4,Th9;
    PProJ(a,b,x,l*y) = ProJ(a,b,p)*ProJ(p,a,x)*ProJ (x,p,l*y) by A1,A2,A5,A6
,Def3;
    then
A7: PProJ(a,b,x,l*y) = (l*ProJ(x,p,y))*(ProJ(a,b,p)*ProJ(p,a,x)) by A6,Th15;
    PProJ(a,b,x,y) = ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,y) by A1,A2,A5,A6,Def3;
    hence thesis by A7,GROUP_1:def 3;
  end;
  now
    assume
A8: x _|_ y;
    then y _|_ x by Th2;
    then l*y _|_ x by Def1;
    then
A9: PProJ(a,b,x,l*y) = 0F by A1,A2,Th33;
    y _|_ x by A8,Th2;
    then l*PProJ(a,b,x,y) = l*0F by A1,A2,Th33;
    hence thesis by A9;
  end;
  hence thesis by A3;
end;
