reserve FS for non empty doubleLoopStr;
reserve F for Field;
reserve R for Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr,
  x, y, z for Scalar of R;
reserve SF for Skew-Field,
  x, y, z for Scalar of SF;
reserve R, R1, R2 for Ring;

theorem Th26:
  BiModule(R1,R2) is BiMod of R1,R2
proof
  for x,y being Scalar of R1, p,q being Scalar of R2, v,w being Vector of
BiModule(R1,R2) holds x*(v+w) = x*v+x*w & (x+y)*v = x*v+y*v & (x*y)*v = x*(y*v)
& (1_R1)*v = v & (v+w)*p = v*p+w*p & v*(p+q) = v*p+v*q & v*(q*p) = (v*q)*p & v*
  (1_R2) = v & x*(v*p) = (x*v)*p by Lm9;
  hence thesis by Def8,Def9,VECTSP_1:def 14,def 15,def 16,def 17;
end;
