
theorem
  for n being Element of NAT st n>0 holds FTSC1 n is symmetric
proof
  let n be Element of NAT;
  set f=Nbdc1 n;
  assume n>0;
  then
A1: FTSC1 n=RelStr(# Seg n,Nbdc1 n #) by Def6;
  let x, y be Element of FTSC1 n;
  x in Seg n by A1;
  then reconsider i=x as Element of NAT;
A2: 1<=i by A1,FINSEQ_1:1;
A3: i=1 & i<n implies Im(f,i)={i,n,i+1} by A1,Def5;
A4: i=1 & i=n implies Im(f,i)={i} by A1,Def5;
A5: i<=n by A1,FINSEQ_1:1;
  y in Seg n by A1;
  then reconsider j=y as Element of NAT;
A6: 1<=j by A1,FINSEQ_1:1;
A7: 1<i & i=n implies Im(f,i)={i,i-'1,1} by A1,Def5;
A8: 1<i & i<n implies Im(f,i)={i,i-'1,i+1} by A1,Def5;
  assume
A9: y in U_FT x;
A10: j<=n by A1,FINSEQ_1:1;
  per cases by A2,A5,XXREAL_0:1;
  suppose
A11: 1<i & i<n;
A12: now
      i-1>0 by A11,XREAL_1:50;
      then
A13:  i-'1=i-1 by XREAL_0:def 2;
      assume
A14:  y=i-'1;
      per cases by A14,A13;
      suppose
A15:    x=j;
        then Im(the InternalRel of FTSC1 n,y)={j,j-'1,j+1} by A1,A11,Def5;
        hence thesis by A15,ENUMSET1:def 1;
      end;
      suppose
A16:    x=j-'1;
        then
A17:    i=(i-'1)-'1 by A14;
        now
          assume i<>0;
          then
A18:      i>=0+1 by NAT_1:13;
          then i-1>=1-1 by XREAL_1:9;
          then
A19:      i-1=i-'1 by XREAL_0:def 2;
          now
            assume
A20:        i=1;
            then i-'1-1<0 by A19;
            hence contradiction by A14,A16,A20,XREAL_0:def 2;
          end;
          then i>1 by A18,XXREAL_0:1;
          then i-1>1-1 by XREAL_1:9;
          then i-'1>=0+1 by A19,NAT_1:13;
          then i-'1-1>=0 by XREAL_1:48;
          then i-'1-'1=i-'1-1 by XREAL_0:def 2;
          hence contradiction by A17,A19;
        end;
        hence thesis by A11;
      end;
      suppose
A21:    x=j+1;
        then
A22:    j<n by A5,NAT_1:13;
A23:    now
          assume j<>1;
          then j>1 by A6,XXREAL_0:1;
          then Im(the InternalRel of FTSC1 n,y)={j,j-'1,j+1} by A1,A22,Def5;
          hence thesis by A21,ENUMSET1:def 1;
        end;
        now
          assume j=1;
          then Im(the InternalRel of FTSC1 n,y)={j,n,j+1} by A1,A22,Def5;
          hence thesis by A21,ENUMSET1:def 1;
        end;
        hence thesis by A23;
      end;
    end;
A24: now
      assume
A25:  y=i+1;
      then
A26:  j-1=x;
      now
        per cases by A11,A26,XREAL_0:def 2;
        case
A27:      x=j;
          then Im(the InternalRel of FTSC1 n,y)={j,j-'1,j+1} by A1,A11,Def5;
          hence thesis by A27,ENUMSET1:def 1;
        end;
        case
A28:      x=j-'1;
          now
            assume j=1;
            then j-1=0;
            hence contradiction by A11,A28,XREAL_0:def 2;
          end;
          then
A29:      j>1 by A6,XXREAL_0:1;
A30:      now
            assume j<>n;
            then j<n by A10,XXREAL_0:1;
            then Im(the InternalRel of FTSC1 n,y)={j,j-'1,j+1} by A1,A29,Def5;
            hence thesis by A28,ENUMSET1:def 1;
          end;
          now
            assume j=n;
            then Im(the InternalRel of FTSC1 n,y)={j,j-'1,1} by A1,A29,Def5;
            hence thesis by A28,ENUMSET1:def 1;
          end;
          hence thesis by A30;
        end;
        case
          x=j+1;
          hence contradiction by A25;
        end;
      end;
      hence thesis;
    end;
    y=i or y=i-'1 or y=i+1 by A1,A8,A9,A11,ENUMSET1:def 1;
    hence thesis by A9,A12,A24;
  end;
  suppose
A31: i=1 & i<n;
    per cases by A1,A3,A9,A31,ENUMSET1:def 1;
    suppose
      y=i & y<>n;
      hence thesis by A1,A3,A31,ENUMSET1:def 1;
    end;
    suppose
      y=n;
      then Im(the InternalRel of FTSC1 n,y)={j,j-'1,1} by A1,A31,Def5;
      hence thesis by A31,ENUMSET1:def 1;
    end;
    suppose
A32:  y=i+1 & y<>n;
      then j-1=i;
      then
A33:  j-'1=i by XREAL_0:def 2;
      j<n by A10,A32,XXREAL_0:1;
      then Im(the InternalRel of FTSC1 n,y)={j,j-'1,j+1} by A1,A31,A32,Def5;
      hence thesis by A33,ENUMSET1:def 1;
    end;
  end;
  suppose
A34: 1<i & i=n;
    per cases by A1,A7,A9,A34,ENUMSET1:def 1;
    suppose
      y=i & y<>1;
      hence thesis by A1,A7,A34,ENUMSET1:def 1;
    end;
    suppose
      y=1;
      then Im(the InternalRel of FTSC1 n,y)={j,n,j+1} by A1,A34,Def5;
      hence thesis by A34,ENUMSET1:def 1;
    end;
    suppose
A35:  y=i-'1 & y<>1;
      then
A36:  1<j by A6,XXREAL_0:1;
      n-1>0 by A34,XREAL_1:50;
      then
A37:  n-1=n-'1 by XREAL_0:def 2;
      n-1+1=n;
      then j<n by A34,A35,A37,NAT_1:13;
      then Im(the InternalRel of FTSC1 n,y)={j,j-'1,j+1} by A1,A36,Def5;
      hence thesis by A34,A35,A37,ENUMSET1:def 1;
    end;
  end;
  suppose
    i=1 & i=n;
    then j=i by A1,A4,A9,TARSKI:def 1;
    hence thesis by A9;
  end;
end;
