reserve T for non empty RelStr,
  A,B for Subset of T,
  x,x2,y,z for Element of T;

theorem Th10:
  (A^i) /\ (B^i) = (A /\ B)^i
proof
  thus (A^i) /\ (B^i) c= (A /\ B)^i
  proof
    let x be object;
    assume
A1: x in (A^i) /\ (B^i);
    then reconsider px=x as Point of T;
    x in (B^i) by A1,XBOOLE_0:def 4;
    then
A2: U_FT px c= B by FIN_TOPO:7;
    x in (A^i) by A1,XBOOLE_0:def 4;
    then U_FT px c= A by FIN_TOPO:7;
    then U_FT px c= A /\ B by A2,XBOOLE_1:19;
    hence thesis by FIN_TOPO:7;
  end;
  let x be object;
  assume
A3: x in (A /\ B)^i;
  then reconsider px=x as Point of T;
A4: U_FT px c= (A /\ B) by A3,FIN_TOPO:7;
  then U_FT px c= B by XBOOLE_1:18;
  then
A5: x in B^i by FIN_TOPO:7;
  U_FT px c= A by A4,XBOOLE_1:18;
  then x in A^i by FIN_TOPO:7;
  hence thesis by A5,XBOOLE_0:def 4;
end;
