let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for M being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF
for W1, W2 being Subspace of M
for v being Element of M st M is_the_direct_sum_of W1,W2 holds
(v |-- W1,W2) `2 = (v |-- W2,W1) `1
let M be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for W1, W2 being Subspace of M
for v being Element of M st M is_the_direct_sum_of W1,W2 holds
(v |-- W1,W2) `2 = (v |-- W2,W1) `1
let W1, W2 be Subspace of M; :: thesis: for v being Element of M st M is_the_direct_sum_of W1,W2 holds
(v |-- W1,W2) `2 = (v |-- W2,W1) `1
let v be Element of M; :: thesis: ( M is_the_direct_sum_of W1,W2 implies (v |-- W1,W2) `2 = (v |-- W2,W1) `1 )
assume A1: M is_the_direct_sum_of W1,W2 ; :: thesis: (v |-- W1,W2) `2 = (v |-- W2,W1) `1
then A2: ( v = ((v |-- W1,W2) `1 ) + ((v |-- W1,W2) `2 ) & (v |-- W1,W2) `1 in W1 & (v |-- W1,W2) `2 in W2 ) by Def6;
A3: M is_the_direct_sum_of W2,W1 by A1, Lm17;
then A4: v = ((v |-- W2,W1) `2 ) + ((v |-- W2,W1) `1 ) by Def6;
( (v |-- W2,W1) `1 in W2 & (v |-- W2,W1) `2 in W1 ) by A3, Def6;
hence (v |-- W1,W2) `2 = (v |-- W2,W1) `1 by A1, A2, A4, Th58; :: thesis: verum