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 Th31:
  not a _|_ p & not x _|_ p & not y _|_ p implies ProJ(p,a,x)*ProJ
  (x,p,y) = (-ProJ(p,a,y))*ProJ(y,p,x)
proof
  set 0F = 0.F, 1F = 1_F;
  assume that
A1: not a _|_ p and
A2: not x _|_ p and
A3: not y _|_ p;
A4: not p _|_ y by A3,Th2;
A5: not p _|_ x by A2,Th2;
A6: now
A7: ProJ(p,a,x) <> 0F by A1,A2,Th23;
    assume
A8: not y _|_ x;
    then
A9: not x _|_ 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)*(1F) = ((-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) <> 0F by A4,A9,Th23;
    then ProJ(y,p,x)*ProJ(p,a,y) = ((-ProJ(x,y,p)")*ProJ(p,a,x))*(1F) 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 thesis by A5,A8,Th25;
  end;
  now
    assume
A11: y _|_ x;
    then ProJ(x,p,y) = 0F by A5,Th23;
    then
A12: ProJ(p,a,x)*ProJ(x,p,y) = 0F;
    x _|_ y by A11,Th2;
    then ProJ(y,p,x) = 0F by A4,Th23;
    hence thesis by A12;
  end;
  hence thesis by A6;
end;
