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 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;
