let V be RealLinearSpace; :: thesis: for W1, W2 being Subspace 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 Subspace 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 Def6;

A3: V is_the_direct_sum_of W2,W1 by A1, Lm16;

then A4: ( v = ((v |-- (W2,W1)) `2) + ((v |-- (W2,W1)) `1) & (v |-- (W2,W1)) `1 in W2 ) by Def6;

A5: (v |-- (W2,W1)) `2 in W1 by A3, Def6;

( v = ((v |-- (W1,W2)) `1) + ((v |-- (W1,W2)) `2) & (v |-- (W1,W2)) `1 in W1 ) by A1, Def6;

hence (v |-- (W1,W2)) `2 = (v |-- (W2,W1)) `1 by A1, A2, A4, A5, Th45; :: thesis: verum

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 Subspace 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 Def6;

A3: V is_the_direct_sum_of W2,W1 by A1, Lm16;

then A4: ( v = ((v |-- (W2,W1)) `2) + ((v |-- (W2,W1)) `1) & (v |-- (W2,W1)) `1 in W2 ) by Def6;

A5: (v |-- (W2,W1)) `2 in W1 by A3, Def6;

( v = ((v |-- (W1,W2)) `1) + ((v |-- (W1,W2)) `2) & (v |-- (W1,W2)) `1 in W1 ) by A1, Def6;

hence (v |-- (W1,W2)) `2 = (v |-- (W2,W1)) `1 by A1, A2, A4, A5, Th45; :: thesis: verum