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 ) ) by RLVECT_1:1, Th1; :: 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 #) )

assume A1: 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 #)

( 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 ) ) by RLVECT_1:1, Th1; :: 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 #) )

assume A1: 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 :: thesis: for u being VECTOR of V holds u in W1 + W2

hence
W1 + W2 = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
by Lm12; :: thesis: verumlet 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 A1;

hence u in W1 + W2 by Th1; :: thesis: verum

end;ex v1, v2 being VECTOR of V st

( v1 in W1 & v2 in W2 & u = v1 + v2 ) by A1;

hence u in W1 + W2 by Th1; :: thesis: verum