let F be Field; for S being OrtSp of F
for b, a, p, c being Element of S st not a _|_ & b _|_ & c _|_ holds
ProJ (a + p),b,c = ProJ a,b,c
let S be OrtSp of F; for b, a, p, c being Element of S st not a _|_ & b _|_ & c _|_ holds
ProJ (a + p),b,c = ProJ a,b,c
let b, a, p, c be Element of S; ( not a _|_ & b _|_ & c _|_ implies ProJ (a + p),b,c = ProJ a,b,c )
assume that
A1:
not a _|_
and
A2:
b _|_
and
A3:
c _|_
; ProJ (a + p),b,c = ProJ a,b,c
p _|_
by A2, Th12;
then
p _|_
by Def2;
then A4:
p _|_
by Th16;
p _|_
by A3, Th12;
then
p _|_
by A4, Def2;
then A5:
c - ((ProJ a,b,c) * b) _|_
by Th12;
a _|_
by A1, Th24;
then
c - ((ProJ a,b,c) * b) _|_
by Th12;
then
c - ((ProJ a,b,c) * b) _|_
by A5, Def2;
then A6:
a + p _|_
by Th12;
not b _|_
by A1, Th12;
then
not b _|_
by A2, Th14;
then A7:
not a + p _|_
by Th12;
then
a + p _|_
by Th24;
hence
ProJ (a + p),b,c = ProJ a,b,c
by A7, A6, Th20; verum