
theorem
  for n being Nat st n>0 holds FTSL1 n is symmetric
proof
  let n be Nat;
  assume n>0;
  then
A1: FTSL1 n=RelStr(# Seg n,Nbdl1 n #) by Def4;
  let x, y be Element of FTSL1 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<=n by A1,FINSEQ_1:1;
  y in Seg n by A1;
  then reconsider j=y as Element of NAT;
A4: U_FT y= {j,max(j-'1,1),min(j+1,n)} by A1,Def3;
A5: U_FT x= {i,max(i-'1,1),min(i+1,n)} by A1,Def3;
  now
A6: now
      assume
A7:   y=max(i-'1,1);
      now
        per cases;
        case
A8:       i-'1>=1;
          then
A9:       y=i-'1 by A7,XXREAL_0:def 10;
          now
            per cases;
            case
              i-1>=0;
              then j=i-1 by A9,XREAL_0:def 2;
              hence x =j or x=max(j-'1,1) or x= min(j+1,n) by A3,XXREAL_0:def 9
;
            end;
            case
              i-1<0;
              hence contradiction by A8,XREAL_0:def 2;
            end;
          end;
          hence x =j or x=max(j-'1,1) or x= min(j+1,n);
        end;
        case
A10:      i-'1<1;
A11:      now
            assume i>1;
            then
A12:        i-1>0 by XREAL_1:50;
            then i-'1=i-1 by XREAL_0:def 2;
            then i-'1>=0+1 by A12,NAT_1:13;
            hence contradiction by A10;
          end;
          y=1 by A7,A10,XXREAL_0:def 10;
          hence x =j or x=max(j-'1,1) or x= min(j+1,n) by A2,A11,XXREAL_0:1;
        end;
      end;
      hence x in U_FT y by A4,ENUMSET1:def 1;
    end;
    assume
A13: y in U_FT x;
A14: now
      assume y=min(i+1,n);
      now
        per cases by A5,A13,ENUMSET1:def 1;
        case
          y=i;
          hence x =j or x=max(j-'1,1) or x= min(j+1,n);
        end;
        case
A15:      y=max(i-'1,1);
          now
            per cases;
            case
A16:          i-'1>=1;
              then
A17:          y=i-'1 by A15,XXREAL_0:def 10;
              now
                per cases;
                case
                  i-1>=0;
                  then j=i-1 by A17,XREAL_0:def 2;
                  hence x =j or x=max(j-'1,1) or x= min(j+1,n) by A3,
XXREAL_0:def 9;
                end;
                case
                  i-1<0;
                  hence contradiction by A16,XREAL_0:def 2;
                end;
              end;
              hence x =j or x=max(j-'1,1) or x= min(j+1,n);
            end;
            case
A18:          i-'1<1;
A19:          now
                assume i>1;
                then
A20:            i-1>0 by XREAL_1:50;
                then i-'1=i-1 by XREAL_0:def 2;
                then i-'1>=0+1 by A20,NAT_1:13;
                hence contradiction by A18;
              end;
              y=1 by A15,A18,XXREAL_0:def 10;
              hence x =j or x=max(j-'1,1) or x= min(j+1,n) by A2,A19,XXREAL_0:1
;
            end;
          end;
          hence x =j or x=max(j-'1,1) or x= min(j+1,n);
        end;
        case
A21:      y=min(i+1,n);
          now
            per cases;
            case
              i+1<=n;
              then
A22:          y=i+1 by A21,XXREAL_0:def 9;
              then
A23:          j-1=j-'1 by XREAL_0:def 2;
              now
                per cases;
                case
                  j-1>=1;
                  hence x =j or x=max(j-'1,1) or x= min(j+1,n) by A22,A23,
XXREAL_0:def 10;
                end;
                case
                  j-1<1;
                  hence contradiction by A1,A22,FINSEQ_1:1;
                end;
              end;
              hence x =j or x=max(j-'1,1) or x= min(j+1,n);
            end;
            case
A24:          i+1>n;
              then y=n by A21,XXREAL_0:def 9;
              then j+1>n by NAT_1:13;
              then
A25:          min(j+1,n)=n by XXREAL_0:def 9;
              i>=n by A24,NAT_1:13;
              hence x =j or x=max(j-'1,1) or x= min(j+1,n) by A3,A25,XXREAL_0:1
;
            end;
          end;
          hence x =j or x=max(j-'1,1) or x= min(j+1,n);
        end;
      end;
      hence x in U_FT y by A4,ENUMSET1:def 1;
    end;
    y=i or y=max(i-'1,1) or y=min(i+1,n) by A5,A13,ENUMSET1:def 1;
    hence x in U_FT y by A13,A6,A14;
  end;
  hence thesis;
end;
