let F be Field; for S being SymSp of F
for a, b, c, p being Element of S st not a _|_ & a _|_ holds
( ProJ (a,(b + p),c) = ProJ (a,b,c) & ProJ (a,b,(c + p)) = ProJ (a,b,c) )
let S be SymSp of F; for a, b, c, p being Element of S st not a _|_ & a _|_ holds
( ProJ (a,(b + p),c) = ProJ (a,b,c) & ProJ (a,b,(c + p)) = ProJ (a,b,c) )
let a, b, c, p be Element of S; ( not a _|_ & a _|_ implies ( ProJ (a,(b + p),c) = ProJ (a,b,c) & ProJ (a,b,(c + p)) = ProJ (a,b,c) ) )
set 0V = 0. S;
assume that
A1:
not a _|_
and
A2:
a _|_
; ( ProJ (a,(b + p),c) = ProJ (a,b,c) & ProJ (a,b,(c + p)) = ProJ (a,b,c) )
not a _|_
by A1, A2, Th4;
then
a _|_
by Th14;
then
a _|_
by VECTSP_1:def 14;
then A3:
a _|_
by VECTSP_1:17;
( a _|_ & a _|_ )
by A1, A2, Th6, Th14;
then
a _|_
by Def1;
then
a _|_
by RLVECT_1:def 3;
then
a _|_
by RLVECT_1:def 3;
then
a _|_
by RLVECT_1:5;
then A4:
a _|_
by RLVECT_1:4;
a _|_
by A2, Def1;
then
a _|_
by A3, Def1;
then
a _|_
by RLVECT_1:def 3;
then
a _|_
by RLVECT_1:5;
then A5:
a _|_
by RLVECT_1:4;
a _|_
by A1, Th14;
hence
( ProJ (a,(b + p),c) = ProJ (a,b,c) & ProJ (a,b,(c + p)) = ProJ (a,b,c) )
by A1, A5, A4, Th12; verum