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 |-- (W1,W2)) `2 in W2 by Def5;
A3: V is_the_direct_sum_of W2,W1 by A1, Th38;
then A4: ( v = ((v |-- (W2,W1)) `2) + ((v |-- (W2,W1)) `1) & (v |-- (W2,W1)) `1 in W2 ) by Def5;
A5: (v |-- (W2,W1)) `2 in W1 by A3, Def5;
( v = ((v |-- (W1,W2)) `1) + ((v |-- (W1,W2)) `2) & (v |-- (W1,W2)) `1 in W1 ) by A1, Def5;
hence (v |-- (W1,W2)) `2 = (v |-- (W2,W1)) `1 by A1, A2, A4, A5, Th43; :: thesis: verum