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 Th25:
  not b _|_ a & not c _|_ a implies ProJ(a,b,c) = ProJ(a,c,b)"
proof
  set 1F = 1_F, 0F = 0.F;
  assume that
A1: not b _|_ a and
A2: not c _|_ a;
A3: ProJ(a,c,b) <> 0F by A1,A2,Th23;
  ProJ(a,b,b)*ProJ(a,b,c)" = ProJ(a,c,b) by A1,A2,Th24;
  then ProJ(a,b,c)"*1F = ProJ(a,c,b) by A1,Th22;
  then
A4: ProJ(a,b,c)" = ProJ(a,c,b);
  ProJ(a,b,c) <> 0F by A1,A2,Th23;
  then 1F = ProJ(a,c,b)*ProJ(a,b,c) by A4,VECTSP_1:def 10;
  then ProJ(a,c,b)" = ProJ(a,c,b)"*(ProJ(a,c,b)*ProJ(a,b,c))
    .= (ProJ(a,c,b)"*ProJ(a,c,b))*ProJ(a,b,c) by GROUP_1:def 3
    .= ProJ(a,b,c)*1F by A3,VECTSP_1:def 10;
  hence thesis;
end;
