let K be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive 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
v |-- W2,W1 = [v2,v1]
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
v |-- W2,W1 = [v2,v1]
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
v |-- W2,W1 = [v2,v1] )
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
v |-- W2,W1 = [v2,v1]
let v, v1, v2 be Vector of V; :: thesis: ( v |-- W1,W2 = [v1,v2] implies v |-- W2,W1 = [v2,v1] )
assume A2: v |-- W1,W2 = [v1,v2] ; :: thesis: v |-- W2,W1 = [v2,v1]
then A3: (v |-- W1,W2) `1 = v1 by MCART_1:7;
then A4: v1 in W1 by A1, VECTSP_5:def 6;
A5: (v |-- W1,W2) `2 = v2 by A2, MCART_1:7;
then A6: v2 in W2 by A1, VECTSP_5:def 6;
v = v2 + v1 by A1, A3, A5, VECTSP_5:def 6;
hence v |-- W2,W1 = [v2,v1] by A1, A4, A6, Th6, VECTSP_5:51; :: thesis: verum