let F be Field; :: thesis: for S being OrtSp of F
for b, a, x, y, z being Element of S st not a _|_ holds
PProJ a,b,x,(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z)
let S be OrtSp of F; :: thesis: for b, a, x, y, z being Element of S st not a _|_ holds
PProJ a,b,x,(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z)
let b, a, x, y, z be Element of S; :: thesis: ( not a _|_ implies PProJ a,b,x,(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z) )
set 0F = 0. F;
assume A1: not a _|_ ; :: thesis: PProJ a,b,x,(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z)

A2: now

assume A3: x <> 0. S ; :: thesis: PProJ a,b,x,(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z)
a <> 0. S by A1, Th11, Th12;
then ex p being Element of S st
( not a _|_ & not x _|_ & not a _|_ & not x _|_ ) by A3, Def2;
then consider p being Element of S such that
A4: ( not a _|_ & not 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, Def7;
( 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, Def7, Th26;
hence PProJ a,b,x,(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z) by A5, VECTSP_1:def 18; :: thesis: verum

end;

now

assume A6: x = 0. S ; :: thesis: PProJ a,b,x,(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z)
then A7: PProJ a,b,x,z = 0. F by A1, Th43;
( PProJ a,b,x,(y + z) = 0. F & PProJ a,b,x,y = 0. F ) by A1, A6, Th43;
hence PProJ a,b,x,(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z) by A7, RLVECT_1:10; :: thesis: verum

end;

hence PProJ a,b,x,(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z) by A2; :: thesis: verum