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 Th8:
  not a9 _|_ a & a9 _|_ b & not b9 _|_ b & b9 _|_ a implies not a9
  +b9 _|_ a & not a9+b9 _|_ b
proof
  set 0V = 0.S;
  assume that
A1: not a9 _|_ a and
A2: a9 _|_ b and
A3: not b9 _|_ b and
A4: b9 _|_ a;
  assume not thesis;
  then
  a9+b9 _|_ a & (-(1_F))*b9 _|_ a or a9+b9 _|_ b & (-(1_F))*a9 _|_ b by A2,A4
,Def1;
  then a9+b9 _|_ a & -b9 _|_ a or a9+b9 _|_ b & -a9 _|_ b by VECTSP_1:14;
  then (a9+b9)+(-b9) _|_ a or -a9+(a9+b9) _|_ b by Def1;
  then a9+(b9+(-b9)) _|_ a or (-a9+a9)+b9 _|_ b by RLVECT_1:def 3;
  then a9+0V _|_ a or 0V+b9 _|_ b by RLVECT_1:5;
  hence contradiction by A1,A3,RLVECT_1:4;
end;
