reserve V for RealLinearSpace;
reserve W,W1,W2,W3 for Subspace of V;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a,a1,a2 for Real;
reserve X,Y,x,y,y1,y2 for set;
reserve C for Coset of W;
reserve C1 for Coset of W1;
reserve C2 for Coset of W2;
reserve t1,t2 for Element of [:the carrier of V, the carrier of V:];

theorem
  for V being RealLinearSpace, W being Subspace of V, L being
Linear_Compl of W, v being VECTOR 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 V be RealLinearSpace, W be Subspace of V, L be Linear_Compl of W;
  V is_the_direct_sum_of W,L by Th35;
  hence thesis by Def6;
end;
