reserve p,p1,p2,p3,p11,p22,q,q1,q2 for Point of TOP-REAL 2,
  f,h for FinSequence of TOP-REAL 2,
  r,r1,r2,s,s1,s2 for Real,
  u,u1,u2,u5 for Point of Euclid 2,
  n,m,i,j,k for Nat,
  N for Nat,
  x,y,z for set;
reserve lambda for Real;

theorem
  i>2 & i in dom f & f is being_S-Seq implies f|i is being_S-Seq
proof
  assume that
A1: i>2 and
A2: i in dom f and
A3: f is being_S-Seq;
A4: f|i = f|Seg i by FINSEQ_1:def 16;
  then
A5: dom(f|i)=dom f /\ Seg i by RELAT_1:61;
  thus f|i is one-to-one
  proof
A6: f is one-to-one by A3;
    let x,y be object;
    assume that
A7: x in dom (f|i) and
A8: y in dom (f|i) and
A9: (f|i).x = (f|i).y;
A10: x in dom f & y in dom f by A5,A7,A8,XBOOLE_0:def 4;
    f.x = (f|i).x by A4,A7,FUNCT_1:47
      .= f.y by A4,A8,A9,FUNCT_1:47;
    hence thesis by A6,A10;
  end;
A11: i<=len f by A2,FINSEQ_3:25;
A12: i in NAT by ORDINAL1:def 12;
  Seg len f = dom f by FINSEQ_1:def 3;
  then Seg i c= dom f by A11,FINSEQ_1:5;
  then
A13: dom (f|i) = Seg i by A5,XBOOLE_1:28;
  hence len (f|i) >= 2 by A1,A12,FINSEQ_1:def 3;
A14: f is unfolded by A3;
  thus f|i is unfolded
  proof
    let j be Nat such that
A15: 1 <= j and
A16: j + 2 <= len (f|i);
    j + 1 <= j + 2 by XREAL_1:6;
    then
A17: j+1 <= len(f|i) by A16,XXREAL_0:2;
    len(f|i) <= len f by A11,A13,A12,FINSEQ_1:def 3;
    then
A18: j + 2 <= len f by A16,XXREAL_0:2;
    1<=j+1 by NAT_1:12;
    then
A19: j+1 in dom(f|i) by A17,FINSEQ_3:25;
    1<=j+1+1 by NAT_1:12;
    then j+1+1 in dom(f|i) by A16,FINSEQ_3:25;
    then
A20: LSeg(f,j+1) = LSeg(f|i,j+1) by A19,Th17;
    j <= j+2 by NAT_1:11;
    then j <= len(f|i) by A16,XXREAL_0:2;
    then j in dom(f|i) by A15,FINSEQ_3:25;
    then LSeg(f,j) = LSeg(f|i,j) by A19,Th17;
    then LSeg(f|i,j) /\ LSeg(f|i,j+1) = {f/.(j+1)} by A14,A15,A18,A20;
    hence thesis by A19,FINSEQ_4:70;
  end;
A21: f is s.n.c. by A3;
  thus f|i is s.n.c.
  proof
    let n,k be Nat;
A22: k+1 >= 1 by NAT_1:11;
A23: n+1 >= 1 by NAT_1:11;
    assume n+1 < k;
    then LSeg(f,n) misses LSeg(f,k) by A21;
    then
A24: LSeg(f,n) /\ LSeg(f,k) = {};
A25: k+1 >= k by NAT_1:11;
A26: n+1 >= n by NAT_1:11;
    per cases;
    suppose
A27:  n in dom(f|i);
      now
        per cases;
        suppose
          n+1 in dom(f|i);
          then
A28:      LSeg(f,n)=LSeg(f|i,n) by A27,Th17;
          now
            per cases;
            suppose
A29:          k in dom (f|i);
              now
                per cases;
                suppose
                  k+1 in dom(f|i);
                  hence LSeg(f|i,n) /\ LSeg(f|i,k) = {} by A24,A28,A29,Th17;
                end;
                suppose
                  not k+1 in dom(f|i);
                  then k+1 > len(f|i) by A22,FINSEQ_3:25;
                  then LSeg(f|i,k) = {} by TOPREAL1:def 3;
                  hence LSeg(f|i,n) /\ LSeg(f|i,k) = {};
                end;
              end;
              hence LSeg(f|i,n) /\ LSeg(f|i,k) = {};
            end;
            suppose
              not k in dom (f|i);
              then k < 1 or k > len(f|i) by FINSEQ_3:25;
              then k < 1 or k+1 > len(f|i) by A25,XXREAL_0:2;
              then LSeg(f|i,k) = {} by TOPREAL1:def 3;
              hence LSeg(f|i,n) /\ LSeg(f|i,k) = {};
            end;
          end;
          hence LSeg(f|i,n) /\ LSeg(f|i,k) = {};
        end;
        suppose
          not n+1 in dom(f|i);
          then n+1 > len(f|i) by A23,FINSEQ_3:25;
          then LSeg(f|i,n) = {} by TOPREAL1:def 3;
          hence LSeg(f|i,n) /\ LSeg(f|i,k) = {};
        end;
      end;
      hence LSeg(f|i,n) /\ LSeg(f|i,k) = {};
    end;
    suppose
      not n in dom(f|i);
      then n < 1 or n > len(f|i) by FINSEQ_3:25;
      then n < 1 or n+1 > len(f|i) by A26,XXREAL_0:2;
      then LSeg(f|i,n) = {} by TOPREAL1:def 3;
      hence LSeg(f|i,n) /\ LSeg(f|i,k) = {};
    end;
  end;
  let j be Nat such that
A30: 1 <= j and
A31: j + 1 <= len (f|i);
  len(f|i) <= len f by A11,A13,A12,FINSEQ_1:def 3;
  then
A32: j + 1 <= len f by A31,XXREAL_0:2;
  j <= j + 1 by NAT_1:11;
  then j <= len(f|i) by A31,XXREAL_0:2;
  then j in dom (f|i) by A30,FINSEQ_3:25;
  then
A33: (f|i)/.j=f/.j by FINSEQ_4:70;
  1<=j+1 by NAT_1:12;
  then j+1 in dom(f|i) by A31,FINSEQ_3:25;
  then
A34: (f|i)/.(j+1)=f/.(j+1) by FINSEQ_4:70;
  f is special by A3;
  hence thesis by A30,A32,A33,A34;
end;
