let V be RealLinearSpace; :: thesis: for W1, W2 being Subspace of V holds
( V is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v} )

let W1, W2 be Subspace of V; :: thesis: ( V is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v} )

set VW1 = the carrier of W1;
set VW2 = the carrier of W2;
0. V in W2 by RLSUB_1:17;
then A1: 0. V in the carrier of W2 by STRUCT_0:def 5;
thus ( V is_the_direct_sum_of W1,W2 implies for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v} ) :: thesis: ( ( for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v} ) implies V is_the_direct_sum_of W1,W2 )
proof
assume A2: V is_the_direct_sum_of W1,W2 ; :: thesis: for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v}

then A3: RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) = W1 + W2 ;
let C1 be Coset of W1; :: thesis: for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v}
let C2 be Coset of W2; :: thesis: ex v being VECTOR of V st C1 /\ C2 = {v}
consider v1 being VECTOR of V such that
A4: C1 = v1 + W1 by RLSUB_1:def 6;
v1 in RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by RLVECT_1:1;
then consider v11, v12 being VECTOR of V such that
A5: v11 in W1 and
A6: v12 in W2 and
A7: v1 = v11 + v12 by ;
consider v2 being VECTOR of V such that
A8: C2 = v2 + W2 by RLSUB_1:def 6;
v2 in RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by RLVECT_1:1;
then consider v21, v22 being VECTOR of V such that
A9: v21 in W1 and
A10: v22 in W2 and
A11: v2 = v21 + v22 by ;
take v = v12 + v21; :: thesis: C1 /\ C2 = {v}
{v} = C1 /\ C2
proof
thus A12: {v} c= C1 /\ C2 :: according to XBOOLE_0:def 10 :: thesis: C1 /\ C2 c= {v}
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {v} or x in C1 /\ C2 )
assume x in {v} ; :: thesis: x in C1 /\ C2
then A13: x = v by TARSKI:def 1;
v21 = v2 - v22 by ;
then v21 in C2 by ;
then C2 = v21 + W2 by RLSUB_1:78;
then A14: x in C2 by ;
v12 = v1 - v11 by ;
then v12 in C1 by ;
then C1 = v12 + W1 by RLSUB_1:78;
then x in C1 by ;
hence x in C1 /\ C2 by ; :: thesis: verum
end;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in C1 /\ C2 or x in {v} )
assume A15: x in C1 /\ C2 ; :: thesis: x in {v}
then C1 meets C2 ;
then reconsider C = C1 /\ C2 as Coset of W1 /\ W2 by Th42;
A16: v in {v} by TARSKI:def 1;
W1 /\ W2 = (0). V by A2;
then ex u being VECTOR of V st C = {u} by RLSUB_1:73;
hence x in {v} by ; :: thesis: verum
end;
hence C1 /\ C2 = {v} ; :: thesis: verum
end;
assume A17: for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v} ; :: thesis: V is_the_direct_sum_of W1,W2
A18: the carrier of W2 is Coset of W2 by RLSUB_1:74;
now :: thesis: for u being VECTOR of V holds u in W1 + W2
let u be VECTOR of V; :: thesis: u in W1 + W2
consider C1 being Coset of W1 such that
A19: u in C1 by Lm17;
consider v being VECTOR of V such that
A20: C1 /\ the carrier of W2 = {v} by ;
A21: v in {v} by TARSKI:def 1;
then v in C1 by ;
then consider v1 being VECTOR of V such that
A22: v1 in W1 and
A23: u - v1 = v by ;
v in the carrier of W2 by ;
then A24: v in W2 by STRUCT_0:def 5;
u = v1 + v by ;
hence u in W1 + W2 by ; :: thesis: verum
end;
hence RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) = W1 + W2 by Lm12; :: according to RLSUB_2:def 4 :: thesis: W1 /\ W2 = (0). V
the carrier of W1 is Coset of W1 by RLSUB_1:74;
then consider v being VECTOR of V such that
A25: the carrier of W1 /\ the carrier of W2 = {v} by ;
0. V in W1 by RLSUB_1:17;
then 0. V in the carrier of W1 by STRUCT_0:def 5;
then A26: 0. V in {v} by ;
the carrier of ((0). V) = {(0. V)} by RLSUB_1:def 3
.= the carrier of W1 /\ the carrier of W2 by
.= the carrier of (W1 /\ W2) by Def2 ;
hence W1 /\ W2 = (0). V by RLSUB_1:30; :: thesis: verum