let F be Field; :: thesis: for S being OrtSp of F
for b, a, x, y being Element of S
for l being Element of F st not a _|_ holds
PProJ a,b,x,(l * y) = l * (PProJ a,b,x,y)
let S be OrtSp of F; :: thesis: for b, a, x, y being Element of S
for l being Element of F st not a _|_ holds
PProJ a,b,x,(l * y) = l * (PProJ a,b,x,y)
let b, a, x, y be Element of S; :: thesis: for l being Element of F st not a _|_ holds
PProJ a,b,x,(l * y) = l * (PProJ a,b,x,y)
let l be Element of F; :: thesis: ( not a _|_ implies PProJ a,b,x,(l * y) = l * (PProJ a,b,x,y) )
set 0F = 0. F;
assume A1: not a _|_ ; :: thesis: PProJ a,b,x,(l * y) = l * (PProJ a,b,x,y)

A2: now

assume not y _|_ ; :: thesis: PProJ a,b,x,(l * y) = l * (PProJ a,b,x,y)
then A3: x <> 0. S by Th11;
a <> 0. S by A1, Th11, Th12;
then ex p being Element of S st
( not a _|_ & not x _|_ & not a _|_ & not x _|_ ) by A3, Def2;
then consider p being Element of S such that
A4: not a _|_ and
A5: not x _|_ ;
PProJ a,b,x,(l * y) = ((ProJ a,b,p) * (ProJ p,a,x)) * (ProJ x,p,(l * y)) by A1, A4, A5, Def7;
then A6: PProJ a,b,x,(l * y) = (l * (ProJ x,p,y)) * ((ProJ a,b,p) * (ProJ p,a,x)) by A5, Th25;
PProJ a,b,x,y = ((ProJ a,b,p) * (ProJ p,a,x)) * (ProJ x,p,y) by A1, A4, A5, Def7;
hence PProJ a,b,x,(l * y) = l * (PProJ a,b,x,y) by A6, GROUP_1:def 4; :: thesis: verum

end;

now

assume A7: y _|_ ; :: thesis: PProJ a,b,x,(l * y) = l * (PProJ a,b,x,y)
then x _|_ by Th12;
then x _|_ by Def2;
then A8: PProJ a,b,x,(l * y) = 0. F by A1, Th44;
x _|_ by A7, Th12;
then l * (PProJ a,b,x,y) = l * (0. F) by A1, Th44;
hence PProJ a,b,x,(l * y) = l * (PProJ a,b,x,y) by A8, VECTSP_1:39; :: thesis: verum

end;

hence PProJ a,b,x,(l * y) = l * (PProJ a,b,x,y) by A2; :: thesis: verum