let F be Field; :: thesis: for S being OrtSp of F
for a, b, 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 a, b, 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 a, b, 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 :: thesis: ( not y _|_ implies PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y)) )

assume not y _|_ ; :: thesis: PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y))
then A3: x <> 0. S by Th1;
a <> 0. S by A1, Th1, Th2;
then ex p being Element of S st
( not a _|_ & not x _|_ & not a _|_ & not x _|_ ) by A3, Def1;
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, Def3;
then A6: PProJ (a,b,x,(l * y)) = (l * (ProJ (x,p,y))) * ((ProJ (a,b,p)) * (ProJ (p,a,x))) by A5, Th12;
PProJ (a,b,x,y) = ((ProJ (a,b,p)) * (ProJ (p,a,x))) * (ProJ (x,p,y)) by A1, A4, A5, Def3;
hence PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y)) by A6, GROUP_1:def 3; :: thesis: verum

end;

now :: thesis: ( y _|_ implies PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y)) )

assume A7: y _|_ ; :: thesis: PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y))
then x _|_ by Th2;
then x _|_ by Def1;
then A8: PProJ (a,b,x,(l * y)) = 0. F by A1, Th29;
x _|_ by A7, Th2;
then l * (PProJ (a,b,x,y)) = l * (0. F) by A1, Th29;
hence PProJ (a,b,x,(l * y)) = l * (PProJ (a,b,x,y)) by A8; :: thesis: verum

end;

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