let F be Field; :: thesis: for S being SymSp of F
for a, p, x, y being Element of S st not p _|_ & not p _|_ & not p _|_ holds
(ProJ (p,a,x)) * (ProJ (x,p,y)) = (- (ProJ (p,a,y))) * (ProJ (y,p,x))
let S be SymSp of F; :: thesis: for a, p, x, y being Element of S st not p _|_ & not p _|_ & not p _|_ holds
(ProJ (p,a,x)) * (ProJ (x,p,y)) = (- (ProJ (p,a,y))) * (ProJ (y,p,x))
let a, p, x, y be Element of S; :: thesis: ( not p _|_ & not p _|_ & not p _|_ implies (ProJ (p,a,x)) * (ProJ (x,p,y)) = (- (ProJ (p,a,y))) * (ProJ (y,p,x)) )
set 0F = 0. F;
set 1F = 1_ F;
assume that
A1: not p _|_ and
A2: not p _|_ and
A3: not p _|_ ; :: thesis: (ProJ (p,a,x)) * (ProJ (x,p,y)) = (- (ProJ (p,a,y))) * (ProJ (y,p,x))
A4: not y _|_ by A3, Th2;
A5: not x _|_ by A2, Th2;

A6: now :: thesis: ( not x _|_ implies (ProJ (p,a,x)) * (ProJ (x,p,y)) = (- (ProJ (p,a,y))) * (ProJ (y,p,x)) )

A7: ProJ (p,a,x) <> 0. F by A1, A2, Th23;
assume A8: not x _|_ ; :: thesis: (ProJ (p,a,x)) * (ProJ (x,p,y)) = (- (ProJ (p,a,y))) * (ProJ (y,p,x))
then A9: not y _|_ by Th2;
(ProJ (p,a,y)) * ((ProJ (p,a,x)) ") = ProJ (p,x,y) by A1, A2, Th24;
then ((ProJ (p,a,y)) * ((ProJ (p,a,x)) ")) * (ProJ (p,a,x)) = ((- ((ProJ (x,y,p)) ")) * (ProJ (y,x,p))) * (ProJ (p,a,x)) by A5, A8, Th27;
then (ProJ (p,a,y)) * (((ProJ (p,a,x)) ") * (ProJ (p,a,x))) = ((- ((ProJ (x,y,p)) ")) * (ProJ (y,x,p))) * (ProJ (p,a,x)) by GROUP_1:def 3;
then (ProJ (p,a,y)) * (1_ F) = ((- ((ProJ (x,y,p)) ")) * (ProJ (y,x,p))) * (ProJ (p,a,x)) by A7, VECTSP_1:def 10;
then ProJ (p,a,y) = ((ProJ (y,x,p)) * (- ((ProJ (x,y,p)) "))) * (ProJ (p,a,x)) ;
then ProJ (p,a,y) = (ProJ (y,x,p)) * ((- ((ProJ (x,y,p)) ")) * (ProJ (p,a,x))) by GROUP_1:def 3;
then (ProJ (y,p,x)) * (ProJ (p,a,y)) = (ProJ (y,p,x)) * (((ProJ (y,p,x)) ") * ((- ((ProJ (x,y,p)) ")) * (ProJ (p,a,x)))) by A4, A9, Th25;
then A10: (ProJ (y,p,x)) * (ProJ (p,a,y)) = ((ProJ (y,p,x)) * ((ProJ (y,p,x)) ")) * ((- ((ProJ (x,y,p)) ")) * (ProJ (p,a,x))) by GROUP_1:def 3;
ProJ (y,p,x) <> 0. F by A4, A9, Th23;
then (ProJ (y,p,x)) * (ProJ (p,a,y)) = ((- ((ProJ (x,y,p)) ")) * (ProJ (p,a,x))) * (1_ F) by A10, VECTSP_1:def 10;
then (ProJ (p,a,y)) * (ProJ (y,p,x)) = (- ((ProJ (x,y,p)) ")) * (ProJ (p,a,x)) ;
then - ((ProJ (p,a,y)) * (ProJ (y,p,x))) = - (- (((ProJ (x,y,p)) ") * (ProJ (p,a,x)))) by VECTSP_1:9;
then - ((ProJ (p,a,y)) * (ProJ (y,p,x))) = ((ProJ (x,y,p)) ") * (ProJ (p,a,x)) by RLVECT_1:17;
then (- (ProJ (p,a,y))) * (ProJ (y,p,x)) = ((ProJ (x,y,p)) ") * (ProJ (p,a,x)) by VECTSP_1:9;
hence (ProJ (p,a,x)) * (ProJ (x,p,y)) = (- (ProJ (p,a,y))) * (ProJ (y,p,x)) by A5, A8, Th25; :: thesis: verum

end;

now :: thesis: ( x _|_ implies (ProJ (p,a,x)) * (ProJ (x,p,y)) = (- (ProJ (p,a,y))) * (ProJ (y,p,x)) )

assume A11: x _|_ ; :: thesis: (ProJ (p,a,x)) * (ProJ (x,p,y)) = (- (ProJ (p,a,y))) * (ProJ (y,p,x))
then ProJ (x,p,y) = 0. F by A5, Th23;
then A12: (ProJ (p,a,x)) * (ProJ (x,p,y)) = 0. F ;
y _|_ by A11, Th2;
then ProJ (y,p,x) = 0. F by A4, Th23;
hence (ProJ (p,a,x)) * (ProJ (x,p,y)) = (- (ProJ (p,a,y))) * (ProJ (y,p,x)) by A12; :: thesis: verum

end;

hence (ProJ (p,a,x)) * (ProJ (x,p,y)) = (- (ProJ (p,a,y))) * (ProJ (y,p,x)) by A6; :: thesis: verum