reserve p for Point of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  v, v1,v2 for FinSequence of REAL,
  r,s for Real,
  n,m,i,j,k for Nat,
  x for set;

theorem Th1:
  i in dom f & 2<=len f implies f/.i in L~f
proof
  assume that
A1: i in dom f and
A2: 2<=len f;
A3: 1<=i by A1,FINSEQ_3:25;
A4: i<=len f by A1,FINSEQ_3:25;
  per cases by A4,XXREAL_0:1;
  suppose
A5: i=len f;
    reconsider l=i-1 as Element of NAT by A3,INT_1:5;
    1+1<=i by A2,A5;
    then 1<=l by XREAL_1:19;
    then
A6: f/.(l+1) in LSeg(f,l) by A4,TOPREAL1:21;
    LSeg(f,l) c=L~f by TOPREAL3:19;
    hence thesis by A6;
  end;
  suppose
    i<len f;
    then i+1<=len f by NAT_1:13;
    then
A7: f/.i in LSeg(f,i) by A3,TOPREAL1:21;
    LSeg(f,i) c= L~f by TOPREAL3:19;
    hence thesis by A7;
  end;
end;
