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 Th45:
  V is_the_direct_sum_of W1,W2 & v = v1 + v2 & v = u1 + u2 & v1 in
  W1 & u1 in W1 & v2 in W2 & u2 in W2 implies v1 = u1 & v2 = u2
proof
  reconsider C2 = v1 + W2 as Coset of W2 by RLSUB_1:def 6;
  reconsider C1 = the carrier of W1 as Coset of W1 by RLSUB_1:74;
A1: v1 in C2 by RLSUB_1:43;
  assume V is_the_direct_sum_of W1,W2;
  then consider u being VECTOR of V such that
A2: C1 /\ C2 = {u} by Th43;
  assume that
A3: v = v1 + v2 & v = u1 + u2 and
A4: v1 in W1 and
A5: u1 in W1 and
A6: v2 in W2 & u2 in W2;
A7: v2 - u2 in W2 by A6,RLSUB_1:23;
  v1 in C1 by A4,STRUCT_0:def 5;
  then v1 in C1 /\ C2 by A1,XBOOLE_0:def 4;
  then
A8: v1 = u by A2,TARSKI:def 1;
  u1 = (v1 + v2) - u2 by A3,Lm14
    .= v1 + (v2 - u2) by RLVECT_1:def 3;
  then
A9: u1 in C2 by A7;
  u1 in C1 by A5,STRUCT_0:def 5;
  then
A10: u1 in C1 /\ C2 by A9,XBOOLE_0:def 4;
  hence v1 = u1 by A2,A8,TARSKI:def 1;
  u1 = u by A10,A2,TARSKI:def 1;
  hence thesis by A3,A8,RLVECT_1:8;
end;
