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;
A1: ( the carrier of W1 is Coset of W1 & the carrier of W2 is Coset of W2 ) by RLSUB_1:90;
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}
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
A3: C1 = v1 + W1 by RLSUB_1:def 6;
consider v2 being VECTOR of V such that
A4: C2 = v2 + W2 by RLSUB_1:def 6;
A5: RLSStruct(# the carrier of V,the U2 of V,the addF of V,the Mult of V #) = W1 + W2 by A2, Def4;
v1 in RLSStruct(# the carrier of V,the U2 of V,the addF of V,the Mult of V #) by RLVECT_1:3;
then consider v11, v12 being VECTOR of V such that
A6: v11 in W1 and
A7: v12 in W2 and
A8: v1 = v11 + v12 by A5, Th5;
v2 in RLSStruct(# the carrier of V,the U2 of V,the addF of V,the Mult of V #) by RLVECT_1:3;
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 A5, Th5;
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 set ; :: 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;
v12 = v1 - v11 by A8, Lm14;
then v12 in C1 by A3, A6, RLSUB_1:80;
then C1 = v12 + W1 by RLSUB_1:94;
then A14: x in C1 by A9, A13;
v21 = v2 - v22 by A11, Lm14;
then v21 in C2 by A4, A10, RLSUB_1:80;
then ( C2 = v21 + W2 & v = v21 + v12 ) by RLSUB_1:94;
then x in C2 by A7, A13;
hence x in C1 /\ C2 by A14, XBOOLE_0:def 4; :: thesis: verum

end;

let x be set ; :: 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 by XBOOLE_0:4;
then reconsider C = C1 /\ C2 as Coset of W1 /\ W2 by Th50;
W1 /\ W2 = (0). V by A2, Def4;
then A16: ex u being VECTOR of V st C = {u} by RLSUB_1:89;
v in {v} by TARSKI:def 1;
hence x in {v} by A12, A15, A16, TARSKI:def 1; :: thesis: verum
thus verum ; :: 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

now

let u be VECTOR of V; :: thesis: u in W1 + W2
consider C1 being Coset of W1 such that
A18: u in C1 by Lm17;
consider v being VECTOR of V such that
A19: C1 /\ the carrier of W2 = {v} by A1, A17;
A20: v in {v} by TARSKI:def 1;
then v in the carrier of W2 by A19, XBOOLE_0:def 4;
then A21: v in W2 by STRUCT_0:def 5;
v in C1 by A19, A20, XBOOLE_0:def 4;
then consider v1 being VECTOR of V such that
A22: v1 in W1 and
A23: u - v1 = v by A18, RLSUB_1:96;
u = v1 + v by A23, Lm14;
hence u in W1 + W2 by A21, A22, Th5; :: thesis: verum

end;

hence RLSStruct(# the carrier of V,the U2 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
consider v being VECTOR of V such that
A24: the carrier of W1 /\ the carrier of W2 = {v} by A1, A17;
( 0. V in W1 & 0. V in W2 ) by RLSUB_1:25;
then ( 0. V in the carrier of W1 & 0. V in the carrier of W2 ) by STRUCT_0:def 5;
then A25: 0. V in {v} by A24, XBOOLE_0:def 4;
the carrier of ((0). V) = {(0. V)} by RLSUB_1:def 3
.= the carrier of W1 /\ the carrier of W2 by A24, A25, TARSKI:def 1
.= the carrier of (W1 /\ W2) by Def2 ;
hence W1 /\ W2 = (0). V by RLSUB_1:38; :: thesis: verum