reserve F for Field;
reserve S for SymSp of F;
reserve a,b,c,d,a9,b9,p,q,r,s,x,y,z for Element of S;
reserve k,l for Element of F;

theorem
  1_F+1_F<>0.F & not b _|_ a implies PProJ(a,b,x,y) = -PProJ(a,b,y,x)
proof
  set 0F = 0.F, 1F = 1_F;
  assume that
A1: 1_F+1_F<>0.F and
A2: not b _|_ a;
A3: now
    assume not x _|_ y;
    then
A4: x <> 0.S & y <> 0.S by Th1,Th2;
    a <> 0.S by A2,Th1,Th2;
    then consider r such that
A5: not r _|_ a and
A6: not r _|_ x and
A7: not r _|_ y by A1,A4,Th10;
A8: not y _|_ r by A7,Th2;
    PProJ(a,b,y,x) = ProJ(a,b,r)*ProJ(r,a,y)*ProJ(y,r,x) by A1,A2,A5,A7,Def3;
    then
A9: PProJ(a,b,y,x) = ProJ(a,b,r)*(ProJ(r,a,y)*ProJ (y,r,x)) by GROUP_1:def 3;
    ( not a _|_ r)& not x _|_ r by A5,A6,Th2;
    then PProJ(a,b,y,x) = ProJ(a,b,r)*((-(ProJ(r,a,x)))*ProJ (x,r,y)) by A8,A9
,Th31;
    then PProJ(a,b,y,x) = ((-(ProJ(r,a,x)))*ProJ(a,b,r))*ProJ (x,r,y) by
GROUP_1:def 3;
    then PProJ(a,b,y,x) = (-ProJ(r,a,x)*ProJ(a,b,r))*ProJ (x,r,y) by VECTSP_1:9
;
    then (-1F)*PProJ(a,b,y,x) = (-1F)*(-ProJ(a,b,r)*ProJ(r,a,x)*ProJ (x,r,y))
    by VECTSP_1:9;
    then
    -(PProJ(a,b,y,x)*(1F)) = (-1F)*(-ProJ(a,b,r)*ProJ(r,a,x)*ProJ (x,r,y)
    ) by VECTSP_1:9;
    then -PProJ(a,b,y,x) = (-1F)*(-ProJ(a,b,r)*ProJ(r,a,x)*ProJ (x,r,y));
    then
A10: -PProJ(a,b,y,x) = (ProJ(a,b,r)*ProJ(r,a,x)*ProJ (x,r,y))*(1F) by
VECTSP_1:10;
    PProJ(a,b,x,y) = ProJ(a,b,r)*ProJ(r,a,x)*ProJ(x,r,y) by A1,A2,A5,A6,Def3;
    hence thesis by A10;
  end;
  now
    assume
A11: x _|_ y;
    then (-1F)*PProJ(a,b,y,x) = (-1F)*0F by A1,A2,Th33;
    then -(PProJ (a,b,y,x)*(1F)) = (-1F)*0F by VECTSP_1:9;
    then
A12: -PProJ(a,b,y,x) = (-1F)*0F;
    y _|_ x by A11,Th2;
    then PProJ(a,b,x,y) = 0F by A1,A2,Th33;
    hence thesis by A12;
  end;
  hence thesis by A3;
end;
