
theorem Th42:
  for V being RealUnitarySpace, W1,W2 being Subspace of V, C1
being Coset of W1, C2 being Coset of W2 st C1 meets C2 holds C1 /\ C2 is Coset
  of W1 /\ W2
proof
  let V be RealUnitarySpace;
  let W1,W2 be Subspace of V;
  let C1 be Coset of W1;
  let C2 be Coset of W2;
  set v = the Element of C1 /\ C2;
  set C = C1 /\ C2;
  assume
A1: C1 /\ C2 <> {};
  then reconsider v as Element of V by TARSKI:def 3;
  v in C1 by A1,XBOOLE_0:def 4;
  then
A2: C1 = v + W1 by RUSUB_1:72;
  v in C2 by A1,XBOOLE_0:def 4;
  then
A3: C2 = v + W2 by RUSUB_1:72;
  reconsider v as VECTOR of V;
A4: v + W1 /\ W2 c= C
  proof
    let x be object;
    assume x in v + (W1 /\ W2);
    then consider u being VECTOR of V such that
A5: u in W1 /\ W2 and
A6: x = v + u by Lm16;
    u in W2 by A5,Th3;
    then x in {v + u2 where u2 is VECTOR of V: u2 in W2} by A6;
    then
A7: x in C2 by A3,RUSUB_1:def 4;
    u in W1 by A5,Th3;
    then x in {v + u1 where u1 is VECTOR of V: u1 in W1} by A6;
    then x in C1 by A2,RUSUB_1:def 4;
    hence thesis by A7,XBOOLE_0:def 4;
  end;
  C c= v + W1 /\ W2
  proof
    let x be object;
    assume
A8: x in C;
    then x in C1 by XBOOLE_0:def 4;
    then consider u1 being VECTOR of V such that
A9: u1 in W1 and
A10: x = v + u1 by A2,Lm16;
    x in C2 by A8,XBOOLE_0:def 4;
    then consider u2 being VECTOR of V such that
A11: u2 in W2 and
A12: x = v + u2 by A3,Lm16;
    u1 = u2 by A10,A12,RLVECT_1:8;
    then u1 in W1 /\ W2 by A9,A11,Th3;
    then x in {v + u where u is VECTOR of V : u in W1 /\ W2} by A10;
    hence thesis by RUSUB_1:def 4;
  end;
  then C = v + W1 /\ W2 by A4;
  hence thesis by RUSUB_1:def 5;
end;
