
theorem
  for n,m being non zero Nat holds FTSS2(n,m) is filled
proof
  let n,m be non zero Nat;
  for x being Element of FTSS2(n,m) holds x in U_FT x
  proof
    let x be Element of FTSS2(n,m);
    consider u,y being object such that
A1: u in Seg n and
A2: y in Seg m and
A3: x=[u,y] by ZFMISC_1:def 2;
    reconsider i=u, j=y as Nat by A1,A2;
A4: FTSL1 m = RelStr(# Seg m,Nbdl1 m #) by FINTOPO4:def 4;
    then reconsider pj=j as Element of FTSL1 m by A2;
A5: i in {i} by ZFMISC_1:31;
    FTSL1 m is filled by FINTOPO4:18;
    then j in U_FT pj;
    then x in [:{i}, Im(Nbdl1 m,j):] by A3,A4,A5,ZFMISC_1:def 2;
    then x in [:{i}, Im(Nbdl1 m,j):] \/ [:Im(Nbdl1 n,u), {j}:] by
XBOOLE_0:def 3;
    hence thesis by A3,Def4;
  end;
  hence thesis;
end;
