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 Th11:
  (Omega).M + W = the ModuleStr of M & W + (Omega).M = the ModuleStr of M
proof
  consider W9 being strict Subspace of M such that
A1: the carrier of W9 = the carrier of (Omega).M;
A2: the carrier of W c= the carrier of W9 by A1,VECTSP_4:def 2;
A3: W9 is Subspace of (Omega).M by Lm6;
  W + (Omega).M = W + W9 by A1,Lm5
    .= W9 by A2,Lm3
    .= the ModuleStr of M by A1,A3,VECTSP_4:31;
  hence thesis by Lm1;
end;
