reserve n for Element of NAT,
  V for Subset of TOP-REAL n,
  s,s1,s2,t,t1,t2 for Point of TOP-REAL n,
  C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  a,p ,p1,p2,q,q1,q2 for Point of TOP-REAL 2;

theorem
  lower_bound(proj1.:(east_halfline a)) = a`1
proof
  set X = proj1.:(east_halfline a);
A1: now
    let r be Real;
    assume r in X;
    then consider x being object such that
A2: x in the carrier of TOP-REAL 2 and
A3: x in east_halfline a and
A4: r = proj1.x by FUNCT_2:64;
    reconsider x as Point of TOP-REAL 2 by A2;
    r = x`1 by A4,PSCOMP_1:def 5;
    hence a`1 <= r by A3,TOPREAL1:def 11;
  end;
A5: now
    reconsider r = a`1 as Real;
    let s be Real;
    assume 0 < s;
    then
A6: r + 0 < a`1 + s by XREAL_1:8;
    take r;
    a in east_halfline a & r = proj1.a by PSCOMP_1:def 5,TOPREAL1:38;
    hence r in X by FUNCT_2:35;
    thus r < a`1 + s by A6;
  end;
  X is bounded_below by Th6;
  hence thesis by A1,A5,SEQ_4:def 2;
end;
