let K be non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr ; :: thesis: for V being VectSp of K
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
for v, v1, v2 being Vector of V st v |-- (W1,W2) = [v1,v2] holds
( v1 in W1 & v2 in W2 )
let V be VectSp of K; :: thesis: for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
for v, v1, v2 being Vector of V st v |-- (W1,W2) = [v1,v2] holds
( v1 in W1 & v2 in W2 )
let W1, W2 be Subspace of V; :: thesis: ( V is_the_direct_sum_of W1,W2 implies for v, v1, v2 being Vector of V st v |-- (W1,W2) = [v1,v2] holds
( v1 in W1 & v2 in W2 ) )
assume A1: V is_the_direct_sum_of W1,W2 ; :: thesis: for v, v1, v2 being Vector of V st v |-- (W1,W2) = [v1,v2] holds
( v1 in W1 & v2 in W2 )
let v, v1, v2 be Vector of V; :: thesis: ( v |-- (W1,W2) = [v1,v2] implies ( v1 in W1 & v2 in W2 ) )
assume v |-- (W1,W2) = [v1,v2] ; :: thesis: ( v1 in W1 & v2 in W2 )
then ( (v |-- (W1,W2)) `1 = v1 & (v |-- (W1,W2)) `2 = v2 ) ;
hence ( v1 in W1 & v2 in W2 ) by A1, VECTSP_5:def 6; :: thesis: verum