reserve F for Field;
reserve S for OrtSp of F;
reserve a,b,c,d,p,q,r,x,y,z for Element of S;
reserve k,l for Element of F;

theorem Th9:
  a _|_ a & b _|_ b implies a+b _|_ a-b
proof
  set 0V = 0.S;
  assume that
A1: a _|_ a and
A2: b _|_ b;
  (-1_F)*a _|_ a by A1,Def1;
  then -a _|_ a by VECTSP_1:14;
  then a _|_ -a by Th2;
  then a _|_ 0V+(-a) by RLVECT_1:4;
  then a _|_ (b+(-b))+(-a) by RLVECT_1:5;
  then a _|_ b+(-b-a) by RLVECT_1:def 3;
  then
A3: a _|_ b-(a+b) by VECTSP_1:17;
  b _|_ b+0V by A2,RLVECT_1:4;
  then b _|_ b+(a+(-a)) by RLVECT_1:5;
  then b _|_ (a+b)-a by RLVECT_1:def 3;
  hence thesis by A3,Def1;
end;
