
theorem
  for n,m being non zero Nat holds FTSS2(n,m) is symmetric
proof
  let n,m be non zero Nat;
  for x, y being Element of FTSS2(n,m) holds y in U_FT x implies x in U_FT y
  proof
    let x, y be Element of FTSS2(n,m);
    consider xu,xv being object such that
A1: xu in Seg n and
A2: xv in Seg m and
A3: x=[xu,xv] by ZFMISC_1:def 2;
    reconsider i=xu, j=xv as Nat by A1,A2;
    consider yu,yv being object such that
A4: yu in Seg n and
A5: yv in Seg m and
A6: y=[yu,yv] by ZFMISC_1:def 2;
    reconsider i2=yu, j2=yv as Nat by A4,A5;
A7: FTSL1 m = RelStr(# Seg m,Nbdl1 m #) by FINTOPO4:def 4;
    then reconsider pj=j as Element of FTSL1 m by A2;
A8: FTSL1 n = RelStr(# Seg n,Nbdl1 n #) by FINTOPO4:def 4;
    then reconsider pi=i as Element of FTSL1 n by A1;
    reconsider pj2=j2 as Element of FTSL1 m by A7,A5;
    reconsider pi2=i2 as Element of FTSL1 n by A8,A4;
    assume y in U_FT x;
    then
A9: y in [:{i}, Im(Nbdl1 m,j):] \/ [:Im(Nbdl1 n,i),{j}:] by A3,Def4;
    now
      per cases by A9,XBOOLE_0:def 3;
      case
        y in [:{i}, Im(Nbdl1 m,j):];
        then consider y1,y2 being object such that
A10:    y1 in {i} and
A11:    y2 in Class(Nbdl1 m,j) and
A12:    y=[y1,y2] by ZFMISC_1:def 2;
        y1 = i by A10,TARSKI:def 1;
        then
A13:    i in {i2} by A6,A10,A12,XTUPLE_0:1;
A14:    FTSL1 m is symmetric by FINTOPO4:19;
        pj2 in U_FT pj by A7,A6,A11,A12,XTUPLE_0:1;
        then pj in U_FT pj2 by A14;
        hence x in [:{i2}, Im(Nbdl1 m,j2):] by A3,A7,A13,ZFMISC_1:def 2;
      end;
      case
        y in [:Im(Nbdl1 n,i),{j}:];
        then consider y1,y2 being object such that
A15:    y1 in Class(Nbdl1 n,i) and
A16:    y2 in {j} and
A17:    y=[y1,y2] by ZFMISC_1:def 2;
        y2 = j by A16,TARSKI:def 1;
        then
A18:    j in {j2} by A6,A16,A17,XTUPLE_0:1;
A19:    FTSL1 n is symmetric by FINTOPO4:19;
        pi2 in U_FT pi by A8,A6,A15,A17,XTUPLE_0:1;
        then pi in U_FT pi2 by A19;
        hence x in [:Im(Nbdl1 n,i2), {j2}:] by A3,A8,A18,ZFMISC_1:def 2;
      end;
    end;
    then x in [:{i2}, Im(Nbdl1 m,j2):] \/ [:Im(Nbdl1 n,i2), {j2}:] by
XBOOLE_0:def 3;
    hence thesis by A6,Def4;
  end;
  hence thesis;
end;
