let V be RealLinearSpace; :: thesis: for W1, W2 being Subspace of V holds
( W1 + W2 = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) iff for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )
let W1, W2 be Subspace of V; :: thesis: ( W1 + W2 = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) iff for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )
thus ( W1 + W2 = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) implies for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ) :: thesis: ( ( for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ) implies W1 + W2 = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) )

proof

assume A1: W1 + W2 = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) ; :: thesis: for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 )
let v be VECTOR of V; :: thesis: ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 )
v in RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by RLVECT_1:1;
hence ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) by A1, Th5; :: thesis: verum

end;

assume A2: for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ; :: thesis: W1 + W2 = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)

now

let u be VECTOR of V; :: thesis: u in W1 + W2
ex v1, v2 being VECTOR of V 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 = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by Lm12; :: thesis: verum