
theorem
  for n,m being non zero Nat holds FTSL2(n,m) is filled
proof
  let n,m be non zero Nat;
  for x being Element of FTSL2(n,m) holds x in U_FT x
  proof
    let x be Element of FTSL2(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: FTSL1 n = RelStr(# Seg n,Nbdl1 n #) by FINTOPO4:def 4;
    then reconsider pi=i as Element of FTSL1 n by A1;
    FTSL1 m is filled by FINTOPO4:18;
    then
A6: j in U_FT pj;
    FTSL1 n is filled by FINTOPO4:18;
    then i in U_FT pi;
    then x in [:Im(Nbdl1 n,i), Im(Nbdl1 m,j):] by A3,A4,A5,A6,ZFMISC_1:def 2;
    hence thesis by A3,Def2;
  end;
  hence thesis;
end;
