reserve r1,r2 for Real;
reserve n,i,i1,i2,j for Nat;
reserve D for non empty set;
reserve f for FinSequence of D;

theorem
  for f being FinSequence of TOP-REAL 2, n being Element of NAT holds L~
  (f/^n) c= L~f
proof
  let f be FinSequence of TOP-REAL 2, n be Element of NAT;
  let x be object;
  assume x in L~(f/^n);
  then x in union { LSeg(f/^n,i) : 1 <= i & i+1 <= len (f/^n) } by
TOPREAL1:def 4;
  then consider Y being set such that
A1: x in Y & Y in { LSeg(f/^n,i) : 1 <= i & i+1 <= len (f/^n) } by TARSKI:def 4
;
  consider i such that
A2: Y= LSeg(f/^n,i) and
A3: 1 <= i and
A4: i+1 <= len (f/^n) by A1;
  now
    per cases;
    case
      n<=len f;
      then LSeg(f/^n,i)=LSeg(f,n+i) by A3,SPPOL_2:4;
      then Y c=L~f by A2,TOPREAL3:19;
      hence thesis by A1;
    end;
    case
      n>len f;
      then f/^n=<*>(the carrier of TOP-REAL 2) by RFINSEQ:def 1;
      hence contradiction by A4;
    end;
  end;
  hence thesis;
end;
