theorem Th19:
  for f being FinSequence of TOP-REAL n,i being Nat holds LSeg(f,i ) c= L~f
proof
  let f be FinSequence of TOP-REAL n,i be Nat;
  let x be object such that
A1: x in LSeg(f,i);
  now
    per cases;
    case
      i < 1;
      hence contradiction by A1,TOPREAL1:def 3;
    end;
    case
A2:   i >= 1;
      now
        per cases;
        case
          i+1 > len f;
          hence contradiction by A1,TOPREAL1:def 3;
        end;
        case
A3:       i+1 <= len f;
          set M = {LSeg(f,j) : 1<=j & j+1<=len f};
          LSeg(f,i) in M by A2,A3;
          hence thesis by A1,TARSKI:def 4;
        end;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
