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
x = 0. S
;
:: thesis: PProJ a,b,x,(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z)then
(
PProJ a,
b,
x,
(y + z) = 0. F &
PProJ a,
b,
x,
y = 0. F &
PProJ a,
b,
x,
z = 0. F )
by A1, Th43;
hence
PProJ a,
b,
x,
(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z)
by RLVECT_1:10;
:: thesis: verum end;
now assume
x <> 0. S
;
:: thesis: PProJ a,b,x,(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z)then
(
a <> 0. S &
x <> 0. S )
by A1, Th11, Th12;
then
ex
p being
Element of
S st
( not
a _|_ & not
x _|_ & not
a _|_ & not
x _|_ )
by Def2;
then consider p being
Element of
S such that A3:
( not
a _|_ & not
x _|_ )
;
A4:
PProJ a,
b,
x,
(y + z) = ((ProJ a,b,p) * (ProJ p,a,x)) * (ProJ x,p,(y + z))
by A1, A3, Def7;
A5:
PProJ a,
b,
x,
y = ((ProJ a,b,p) * (ProJ p,a,x)) * (ProJ x,p,y)
by A1, A3, Def7;
A6:
PProJ a,
b,
x,
z = ((ProJ a,b,p) * (ProJ p,a,x)) * (ProJ x,p,z)
by A1, A3, Def7;
ProJ x,
p,
(y + z) = (ProJ x,p,y) + (ProJ x,p,z)
by A3, Th26;
hence
PProJ a,
b,
x,
(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z)
by A4, A5, A6, VECTSP_1:def 18;
:: thesis: verum end;
hence
PProJ a,b,x,(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z)
by A2; :: thesis: verum