let V be RealLinearSpace; :: thesis: for W1, W2 being Subspace of V holds

( W1 is Subspace of W1 + W2 & W2 is Subspace of W1 + W2 )

let W1, W2 be Subspace of V; :: thesis: ( W1 is Subspace of W1 + W2 & W2 is Subspace of W1 + W2 )

the carrier of W1 c= the carrier of (W1 + W2) by Lm2;

hence W1 is Subspace of W1 + W2 by RLSUB_1:28; :: thesis: W2 is Subspace of W1 + W2

the carrier of W2 c= the carrier of (W2 + W1) by Lm2;

then the carrier of W2 c= the carrier of (W1 + W2) by Lm1;

hence W2 is Subspace of W1 + W2 by RLSUB_1:28; :: thesis: verum

( W1 is Subspace of W1 + W2 & W2 is Subspace of W1 + W2 )

let W1, W2 be Subspace of V; :: thesis: ( W1 is Subspace of W1 + W2 & W2 is Subspace of W1 + W2 )

the carrier of W1 c= the carrier of (W1 + W2) by Lm2;

hence W1 is Subspace of W1 + W2 by RLSUB_1:28; :: thesis: W2 is Subspace of W1 + W2

the carrier of W2 c= the carrier of (W2 + W1) by Lm2;

then the carrier of W2 c= the carrier of (W1 + W2) by Lm1;

hence W2 is Subspace of W1 + W2 by RLSUB_1:28; :: thesis: verum