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;
reserve F for Field;
reserve V for VectSp of F;
reserve W for Subspace of V;
reserve W,W1,W2 for Subspace of V;
reserve W1,W2 for Subspace of M;
reserve W for Subspace of V;
reserve W1,W2 for Subspace of M;
reserve u,u1,u2,v for Element of M;
reserve C1 for Coset of W1;
reserve C2 for Coset of W2;
reserve t1,t2 for Element of [:the carrier of M, the carrier of M:];
reserve W for Subspace of V;

theorem
  for L being Linear_Compl of W, v being Element of V, t being Element
  of [:the carrier of V,the carrier of V:] holds t`1 + t`2 = v & t`1 in W & t`2
  in L implies t = v |-- (W,L)
proof
  let L be Linear_Compl of W;
  let v be Element of V;
  let t be Element of [:the carrier of V,the carrier of V:];
  V is_the_direct_sum_of W,L by Th38;
  hence thesis by Def6;
end;
