let V be RealLinearSpace; :: thesis: for W1, W2, W3 being Subspace of V st W1 is Subspace of W3 & W2 is Subspace of W3 holds

W1 + W2 is Subspace of W3

let W1, W2, W3 be Subspace of V; :: thesis: ( W1 is Subspace of W3 & W2 is Subspace of W3 implies W1 + W2 is Subspace of W3 )

assume A1: ( W1 is Subspace of W3 & W2 is Subspace of W3 ) ; :: thesis: W1 + W2 is Subspace of W3

W1 + W2 is Subspace of W3

let W1, W2, W3 be Subspace of V; :: thesis: ( W1 is Subspace of W3 & W2 is Subspace of W3 implies W1 + W2 is Subspace of W3 )

assume A1: ( W1 is Subspace of W3 & W2 is Subspace of W3 ) ; :: thesis: W1 + W2 is Subspace of W3

now :: thesis: for v being VECTOR of V st v in W1 + W2 holds

v in W3

hence
W1 + W2 is Subspace of W3
by RLSUB_1:29; :: thesis: verumv in W3

let v be VECTOR of V; :: thesis: ( v in W1 + W2 implies v in W3 )

assume v in W1 + W2 ; :: thesis: v in W3

then consider v1, v2 being VECTOR of V such that

A2: ( v1 in W1 & v2 in W2 ) and

A3: v = v1 + v2 by RLSUB_2:1;

( v1 in W3 & v2 in W3 ) by A1, A2, RLSUB_1:8;

hence v in W3 by A3, RLSUB_1:20; :: thesis: verum

end;assume v in W1 + W2 ; :: thesis: v in W3

then consider v1, v2 being VECTOR of V such that

A2: ( v1 in W1 & v2 in W2 ) and

A3: v = v1 + v2 by RLSUB_2:1;

( v1 in W3 & v2 in W3 ) by A1, A2, RLSUB_1:8;

hence v in W3 by A3, RLSUB_1:20; :: thesis: verum