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+z) = PProJ(a,b,x,y) + PProJ(a,b,x,z)
proof
  set 0F = 0.F;
  assume
A1: not b _|_ a;
A2: now
    assume
A3: x <> 0.S;
    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)& not p _|_ x;
A5: PProJ(a,b,x,y+z) = ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,y+z) & PProJ(a,b,
    x,y) = ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,y) by A1,A4,Def3;
    PProJ(a,b,x,z) = ProJ(a,b,p)*ProJ(p,a,x)*ProJ(x,p,z) & ProJ(x,p,y+z)
    = ProJ( x,p,y) + ProJ(x,p,z) by A1,A4,Def3,Th13;
    hence thesis by A5,VECTSP_1:def 7;
  end;
  now
    assume
A6: x = 0.S;
    then
A7: PProJ (a,b,x,z) = 0F by A1,Th28;
    PProJ(a,b,x,y+z) = 0F & PProJ(a,b,x,y) = 0F by A1,A6,Th28;
    hence thesis by A7,RLVECT_1:4;
  end;
  hence thesis by A2;
end;
