
theorem
  for n being Element of NAT st n>0 holds FTSC1 n is filled
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 be Element of FTSC1 n;
  x in Seg n by A1;
  then reconsider i=x as Element of NAT;
A2: 1<=i & i<=n by A1,FINSEQ_1:1;
A3: i=1 & i<n implies Im(f,i)={i,n,i+1} by A1,Def5;
A4: 1<i & i<n implies Im(f,i)={i,i-'1,i+1} by A1,Def5;
A5: i=1 & i=n implies Im(f,i)={i} by A1,Def5;
A6: 1<i & i=n implies Im(f,i)={i,i-'1,1} by A1,Def5;
  per cases by A2,XXREAL_0:1;
  suppose
    1<i & i<n;
    hence thesis by A1,A4,ENUMSET1:def 1;
  end;
  suppose
    i=1 & i<n;
    hence thesis by A1,A3,ENUMSET1:def 1;
  end;
  suppose
    1<i & i=n;
    hence thesis by A1,A6,ENUMSET1:def 1;
  end;
  suppose
    i=1 & i=n;
    hence thesis by A1,A5,TARSKI:def 1;
  end;
end;
