reserve V for RealLinearSpace;
reserve W,W1,W2,W3 for Subspace of V;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a,a1,a2 for Real;
reserve X,Y,x,y,y1,y2 for set;
reserve C for Coset of W;
reserve C1 for Coset of W1;
reserve C2 for Coset of W2;

theorem Th43:
  for V being RealLinearSpace, W1,W2 being Subspace of V holds V
  is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1, C2 being Coset of W2
  ex v being VECTOR of V st C1 /\ C2 = {v}
proof
  let V be RealLinearSpace, W1,W2 be Subspace of 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 VW2 by STRUCT_0:def 5;
  thus V is_the_direct_sum_of W1,W2 implies for C1 being Coset of W1, C2 being
  Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v}
  proof
    assume
A2: V is_the_direct_sum_of W1,W2;
    then
A3: the RLSStruct of V = W1 + W2;
    let C1 be Coset of W1, C2 be Coset of W2;
    consider v1 being VECTOR of V such that
A4: C1 = v1 + W1 by RLSUB_1:def 6;
    v1 in the RLSStruct 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 A3,Th1;
    consider v2 being VECTOR of V such that
A8: C2 = v2 + W2 by RLSUB_1:def 6;
    v2 in the RLSStruct 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 A3,Th1;
    take v = v12 + v21;
    {v} = C1 /\ C2
    proof
      thus
A12:  {v} c= C1 /\ C2
      proof
        let x be object;
        assume x in {v};
        then
A13:    x = v by TARSKI:def 1;
        v21 = v2 - v22 by A11,Lm14;
        then v21 in C2 by A8,A10,RLSUB_1:64;
        then C2 = v21 + W2 by RLSUB_1:78;
        then
A14:    x in C2 by A6,A13;
        v12 = v1 - v11 by A7,Lm14;
        then v12 in C1 by A4,A5,RLSUB_1:64;
        then C1 = v12 + W1 by RLSUB_1:78;
        then x in C1 by A9,A13;
        hence thesis by A14,XBOOLE_0:def 4;
      end;
      let x be object;
      assume
A15:  x in C1 /\ C2;
      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 thesis by A12,A15,A16,TARSKI:def 1;
    end;
    hence thesis;
  end;
  assume
A17: for C1 being Coset of W1, C2 being Coset of W2 ex v being VECTOR of
  V st C1 /\ C2 = {v};
A18: VW2 is Coset of W2 by RLSUB_1:74;
  now
    let u be VECTOR of V;
    consider C1 being Coset of W1 such that
A19: u in C1 by Lm17;
    consider v being VECTOR of V such that
A20: C1 /\ VW2 = {v} by A18,A17;
A21: v in {v} by TARSKI:def 1;
    then v in C1 by A20,XBOOLE_0:def 4;
    then consider v1 being VECTOR of V such that
A22: v1 in W1 and
A23: u - v1 = v by A19,RLSUB_1:80;
    v in VW2 by A20,A21,XBOOLE_0:def 4;
    then
A24: v in W2 by STRUCT_0:def 5;
    u = v1 + v by A23,Lm14;
    hence u in W1 + W2 by A24,A22,Th1;
  end;
  hence the RLSStruct of V = W1 + W2 by Lm12;
  VW1 is Coset of W1 by RLSUB_1:74;
  then consider v being VECTOR of V such that
A25: VW1 /\ VW2 = {v} by A18,A17;
  0.V in W1 by RLSUB_1:17;
  then 0.V in VW1 by STRUCT_0:def 5;
  then
A26: 0.V in {v} by A25,A1,XBOOLE_0:def 4;
  the carrier of (0).V = {0.V} by RLSUB_1:def 3
    .= VW1 /\ VW2 by A25,A26,TARSKI:def 1
    .= the carrier of W1 /\ W2 by Def2;
  hence thesis by RLSUB_1:30;
end;
