
theorem
  for n,m being non zero Nat holds FTSL2(n,m) is symmetric
proof
  let n,m be non zero Nat;
  for x, y being Element of FTSL2(n,m) holds y in U_FT x implies x in U_FT y
  proof
A1: FTSL1 m is symmetric by FINTOPO4:19;
    let x, y be Element of FTSL2(n,m);
    consider xu,xv being object such that
A2: xu in Seg n and
A3: xv in Seg m and
A4: x=[xu,xv] by ZFMISC_1:def 2;
    reconsider i=xu, j=xv as Nat by A2,A3;
    consider yu,yv being object such that
A5: yu in Seg n and
A6: yv in Seg m and
A7: y=[yu,yv] by ZFMISC_1:def 2;
    reconsider i2=yu, j2=yv as Nat by A5,A6;
A8: FTSL1 m = RelStr(# Seg m,Nbdl1 m #) by FINTOPO4:def 4;
    then reconsider pj=j as Element of FTSL1 m by A3;
    reconsider pj2=j2 as Element of FTSL1 m by A8,A6;
    assume y in U_FT x;
    then y in [:Im(Nbdl1 n,i), Im(Nbdl1 m,j):] by A4,Def2;
    then
A9: ex y1,y2 being object
st y1 in Class(Nbdl1 n,i) & y2 in Class(Nbdl1 m,j)
    & y=[y1,y2] by ZFMISC_1:def 2;
    then j2 in U_FT pj by A8,A7,XTUPLE_0:1;
    then
A10: j in U_FT pj2 by A1;
A11: FTSL1 n = RelStr(# Seg n,Nbdl1 n #) by FINTOPO4:def 4;
    then reconsider pi=i as Element of FTSL1 n by A2;
A12: FTSL1 n is symmetric by FINTOPO4:19;
    reconsider pi2=i2 as Element of FTSL1 n by A11,A5;
    pi2 in U_FT pi by A11,A7,A9,XTUPLE_0:1;
    then pi in U_FT pi2 by A12;
    then x in [:Im(Nbdl1 n,i2), Im(Nbdl1 m,j2):] by A4,A8,A11,A10,
ZFMISC_1:def 2;
    hence thesis by A7,Def2;
  end;
  hence thesis;
end;
