
theorem Th45:
  for V being RealUnitarySpace, W1,W2 being Subspace of V, v,v1,v2
,u1,u2 being VECTOR of V st 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 holds v1 = u1 & v2 = u2
proof
  let V be RealUnitarySpace;
  let W1,W2 be Subspace of V;
  let v,v1,v2,u1,u2 be VECTOR of V;
  reconsider C2 = v1 + W2 as Coset of W2 by RUSUB_1:def 5;
  reconsider C1 = the carrier of W1 as Coset of W1 by RUSUB_1:68;
A1: v1 in C2 by RUSUB_1:37;
  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,RUSUB_1:17;
  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;
A9: u1 in C1 by A5,STRUCT_0:def 5;
  u1 = (v1 + v2) - u2 by A3,RLSUB_2:61
    .= v1 + (v2 - u2) by RLVECT_1:def 3;
  then u1 in C2 by A7,Lm16;
  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;
