let R be Ring; :: thesis: for V being RightMod of R
for W1, W2 being Submodule of V
for v being Vector of V st V is_the_direct_sum_of W1,W2 holds
(v |-- W1,W2) `2 = (v |-- W2,W1) `1
let V be RightMod of R; :: thesis: for W1, W2 being Submodule of V
for v being Vector of V st V is_the_direct_sum_of W1,W2 holds
(v |-- W1,W2) `2 = (v |-- W2,W1) `1
let W1, W2 be Submodule of V; :: thesis: for v being Vector of V st V is_the_direct_sum_of W1,W2 holds
(v |-- W1,W2) `2 = (v |-- W2,W1) `1
let v be Vector of V; :: thesis: ( V is_the_direct_sum_of W1,W2 implies (v |-- W1,W2) `2 = (v |-- W2,W1) `1 )
assume A1: V 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 Def5;
A3: V is_the_direct_sum_of W2,W1 by A1, Th46;
then A4: v = ((v |-- W2,W1) `2 ) + ((v |-- W2,W1) `1 ) by Def5;
( (v |-- W2,W1) `1 in W2 & (v |-- W2,W1) `2 in W1 ) by A3, Def5;
hence (v |-- W1,W2) `2 = (v |-- W2,W1) `1 by A1, A2, A4, Th51; :: thesis: verum