reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr;
reserve M for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over GF;
reserve W,W1,W2,W3 for Subspace of M;
reserve u,u1,u2,v,v1,v2 for Element of M;
reserve X,Y for set, x,y,y1,y2 for object;

theorem Th2:
  v in W1 or v in W2 implies v in W1 + W2
proof
  assume
A1: v in W1 or v in W2;
  now
    per cases by A1;
    suppose
A2:   v in W1;
      v = v + 0.M & 0.M in W2 by RLVECT_1:4,VECTSP_4:17;
      hence thesis by A2,Th1;
    end;
    suppose
A3:   v in W2;
      v = 0.M + v & 0.M in W1 by RLVECT_1:4,VECTSP_4:17;
      hence thesis by A3,Th1;
    end;
  end;
  hence thesis;
end;
