let V be RealUnitarySpace; :: thesis: for W1, W2 being Subspace of V
for v being VECTOR of V st W1 is Subspace of W2 holds
v + W1 c= v + W2
let W1, W2 be Subspace of V; :: thesis: for v being VECTOR of V st W1 is Subspace of W2 holds
v + W1 c= v + W2
let v be VECTOR of V; :: thesis: ( W1 is Subspace of W2 implies v + W1 c= v + W2 )
assume A1: W1 is Subspace of W2 ; :: thesis: v + W1 c= v + W2
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in v + W1 or x in v + W2 )
assume x in v + W1 ; :: thesis: x in v + W2
then consider u being VECTOR of V such that
A2: u in W1 and
A3: x = v + u by RUSUB_2:63;
u in W2 by A1, A2, RUSUB_1:1;
hence x in v + W2 by A3, RUSUB_2:63; :: thesis: verum