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 Th13:
  1_F+1_F<>0.F implies a _|_ a
proof
  set 1F = 1_F;
  assume
A1: 1_F+1_F<>0.F;
  now
    assume a <> 0.S;
    then consider c such that
A2: not c _|_ a by Def1;
    consider k such that
A3: a-k*c _|_ a by A2,Def1;
    a _|_ a+(-k*c) by A3,Th2;
    then -k*c _|_ a+a by Def1;
    then -k*c _|_ a+(1F)*a;
    then -k*c _|_ (1F)*a+(1F)*a;
    then (-1F)*(k*c) _|_ (1F)*a+(1F)*a by VECTSP_1:14;
    then (-1F)*(k*c) _|_ ((1F)+(1F))*a by VECTSP_1:def 15;
    then ((-1F)*k)*c _|_ ((1F)+(1F))*a by VECTSP_1:def 16;
    then (-(k*(1F)))*c _|_ ((1F)+(1F))*a by VECTSP_1:9;
    then (-k)*c _|_ ((1_F)+(1F))*a;
    then ((1_F)+(1_F))*a _|_ (-k)*c by Th2;
    then ((1_F)+(1_F))"*(((1_F)+(1_F))*a) _|_ (-k)*c by Def1;
    then a _|_ (-k)*c by A1,VECTSP_1:20;
    then
A4: (-k)*c _|_ a by Th2;
    a+(-k)*c _|_ a by A3,VECTSP_1:21;
    hence thesis by A4,Th4;
  end;
  hence thesis by Th1;
end;
