let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for W1, W2 being Subspace of M holds
( W1 + W2 = VectSpStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) iff for v being Element of M ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for W1, W2 being Subspace of M holds
( W1 + W2 = VectSpStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) iff for v being Element of M ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )
let W1, W2 be Subspace of M; :: thesis: ( W1 + W2 = VectSpStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) iff for v being Element of M ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )
thus ( W1 + W2 = VectSpStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) implies for v being Element of M ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ) :: thesis: ( ( for v being Element of M ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ) implies W1 + W2 = VectSpStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) )

proof

assume A1: W1 + W2 = VectSpStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) ; :: thesis: for v being Element of M ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & v = v1 + v2 )
let v be Element of M; :: thesis: ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & v = v1 + v2 )
v in (Omega). M by RLVECT_1:1;
hence ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) by A1, Th5; :: thesis: verum

end;

assume A2: for v being Element of M ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ; :: thesis: W1 + W2 = VectSpStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #)

now

thus W1 + W2 is Subspace of (Omega). M by Lm6; :: thesis: for u being Element of M holds u in W1 + W2
let u be Element of M; :: thesis: u in W1 + W2
ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & u = v1 + v2 ) by A2;
hence u in W1 + W2 by Th5; :: thesis: verum

end;

hence W1 + W2 = VectSpStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) by VECTSP_4:32; :: thesis: verum