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

A2: now

assume not y _|_ ; :: thesis: PProJ (a,b,x,y) = PProJ (a,b,y,x)
then A3: ( x <> 0. S & y <> 0. S ) by Th11, Th12;
a <> 0. S by A1, Th11, Th12;
then ex r being Element of S st
( not a _|_ & not x _|_ & not y _|_ & not a _|_ ) by A3, Def2;
then consider r being Element of S such that
A4: not a _|_ and
A5: not x _|_ and
A6: not y _|_ ;
A7: not r _|_ by A6, Th12;
PProJ (a,b,y,x) = ((ProJ (a,b,r)) * (ProJ (r,a,y))) * (ProJ (y,r,x)) by A1, A4, A6, Def7;
then A8: PProJ (a,b,y,x) = (ProJ (a,b,r)) * ((ProJ (r,a,y)) * (ProJ (y,r,x))) by GROUP_1:def 3;
( not r _|_ & not r _|_ ) by A4, A5, Th12;
then A9: PProJ (a,b,y,x) = (ProJ (a,b,r)) * ((ProJ (r,a,x)) * (ProJ (x,r,y))) by A7, A8, Th40;
PProJ (a,b,x,y) = ((ProJ (a,b,r)) * (ProJ (r,a,x))) * (ProJ (x,r,y)) by A1, A4, A5, Def7;
hence PProJ (a,b,x,y) = PProJ (a,b,y,x) by A9, GROUP_1:def 3; :: thesis: verum

end;

now

assume y _|_ ; :: thesis: PProJ (a,b,x,y) = PProJ (a,b,y,x)
then ( x _|_ & PProJ (a,b,y,x) = 0. F ) by A1, Th12, Th44;
hence PProJ (a,b,x,y) = PProJ (a,b,y,x) by A1, Th44; :: thesis: verum

end;

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