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 Th42:
  C1 meets C2 implies C1 /\ C2 is Coset of W1 /\ W2
proof
  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 C2 by A1,XBOOLE_0:def 4;
  then
A2: C2 = v + W2 by RLSUB_1:78;
  v in C1 by A1,XBOOLE_0:def 4;
  then
A3: C1 = v + W1 by RLSUB_1:78;
  C is Coset of W1 /\ W2
  proof
    take v;
    thus C c= v + W1 /\ W2
    proof
      let x be object;
      assume
A4:   x in C;
      then x in C1 by XBOOLE_0:def 4;
      then consider u1 such that
A5:   u1 in W1 and
A6:   x = v + u1 by A3,RLSUB_1:63;
      x in C2 by A4,XBOOLE_0:def 4;
      then consider u2 such that
A7:   u2 in W2 and
A8:   x = v + u2 by A2,RLSUB_1:63;
      u1 = u2 by A6,A8,RLVECT_1:8;
      then u1 in W1 /\ W2 by A5,A7,Th3;
      hence thesis by A6;
    end;
    let x be object;
    assume x in v + (W1 /\ W2);
    then consider u such that
A9: u in W1 /\ W2 and
A10: x = v + u by RLSUB_1:63;
    u in W2 by A9,Th3;
    then
A11: x in {v + u2 : u2 in W2} by A10;
    u in W1 by A9,Th3;
    then x in {v + u1 : u1 in W1} by A10;
    hence thesis by A3,A2,A11,XBOOLE_0:def 4;
  end;
  hence thesis;
end;
