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 Th4:
  not b _|_ a & c _|_ a implies not b+c _|_ a
proof
  assume that
A1: not b _|_ a and
A2: c _|_ a;
  (-1_F)*c _|_ a by A2,Def1;
  then
A3: -c _|_ a by VECTSP_1:14;
  assume not thesis;
  then (b+c)+(-c) _|_ a by A3,Def1;
  then b+(c+(-c)) _|_ a by RLVECT_1:def 3;
  then b+0.S _|_ a by RLVECT_1:5;
  hence contradiction by A1,RLVECT_1:4;
end;
