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 Th1:
  0.S _|_ a
proof
  set 0V = 0.S, 0F = 0.F;
A1: now
    assume not a _|_ a;
    then consider m being Element of F such that
A2: (0V-m*a) _|_ a by Def1;
    0F*(0V-m*a) _|_ a by A2,Def1;
    hence thesis by VECTSP_1:14;
  end;
  now
    assume a _|_ a;
    then 0F*a _|_ a by Def1;
    hence thesis by VECTSP_1:14;
  end;
  hence thesis by A1;
end;
