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,y+z) = PProJ(a,b,x,y) +
  PProJ (a,b,x,z)
proof
  assume that
A1: 1_F+1_F<>0.F and
A2: not b _|_ a;
A3: now
    assume
A4: x <> 0.S;
    a <> 0.S by A2,Th1,Th2;
    then consider p such that
A5: ( not p _|_ a)& not p _|_ x by A4,Th9;
A6: 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,A2,A5,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,A2,A5,Def3,Th16;
    hence thesis by A6,VECTSP_1:def 7;
  end;
  set 0F = 0.F;
  now
    assume
A7: x = 0.S;
    then
A8: PProJ (a,b,x,z) = 0F by A1,A2,Th32;
    PProJ(a,b,x,y+z) = 0F & PProJ(a,b,x,y) = 0F by A1,A2,A7,Th32;
    hence thesis by A8,RLVECT_1:4;
  end;
  hence thesis by A3;
end;
