let F be Field; :: thesis: for S being SymSp of F
for a, b, p, q being Element of S st not a _|_ & not a _|_ holds
(ProJ (a,b,p)) * ((ProJ (a,b,q)) ") = ProJ (a,q,p)
let S be SymSp of F; :: thesis: for a, b, p, q being Element of S st not a _|_ & not a _|_ holds
(ProJ (a,b,p)) * ((ProJ (a,b,q)) ") = ProJ (a,q,p)
let a, b, p, q be Element of S; :: thesis: ( not a _|_ & not a _|_ implies (ProJ (a,b,p)) * ((ProJ (a,b,q)) ") = ProJ (a,q,p) )
assume that
A1: not a _|_ and
A2: not a _|_ ; :: thesis: (ProJ (a,b,p)) * ((ProJ (a,b,q)) ") = ProJ (a,q,p)
( a _|_ & a _|_ ) by A1, A2, 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 a _|_ by RLVECT_1:4;
then A3: a _|_ by A1, Th15;
a _|_ by A1, Th14;
then (ProJ (a,q,p)) * (ProJ (a,b,q)) = ProJ (a,b,p) by A1, A3, Th12;
then A4: (ProJ (a,q,p)) * ((ProJ (a,b,q)) * ((ProJ (a,b,q)) ")) = (ProJ (a,b,p)) * ((ProJ (a,b,q)) ") by GROUP_1:def 3;
ProJ (a,b,q) <> 0. F by A1, A2, Th23;
then (ProJ (a,q,p)) * (1_ F) = (ProJ (a,b,p)) * ((ProJ (a,b,q)) ") by A4, VECTSP_1:def 10;
hence (ProJ (a,b,p)) * ((ProJ (a,b,q)) ") = ProJ (a,q,p) ; :: thesis: verum