let V be RealUnitarySpace; :: thesis: for W being Subspace of V
for u, v being VECTOR of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 + w2 = v + u
let W be Subspace of V; :: thesis: for u, v being VECTOR of V
for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 + w2 = v + u
let u, v be VECTOR of V; :: thesis: for w1, w2 being VECTOR of W st w1 = v & w2 = u holds
w1 + w2 = v + u
let w1, w2 be VECTOR of W; :: thesis: ( w1 = v & w2 = u implies w1 + w2 = v + u )
assume A1: ( v = w1 & u = w2 ) ; :: thesis: w1 + w2 = v + u
w1 + w2 = ( the addF of V || the carrier of W) . [w1,w2] by Def1;
hence w1 + w2 = v + u by A1, FUNCT_1:49; :: thesis: verum