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 Th8:
  upper_bound(proj2.:(south_halfline a)) = a`2
proof
  set X = proj2.:(south_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 south_halfline a and
A4: r = proj2.x by FUNCT_2:64;
    reconsider x as Point of TOP-REAL 2 by A2;
    r = x`2 by A4,PSCOMP_1:def 6;
    hence r <= a`2 by A3,TOPREAL1:def 12;
  end;
A5: now
    reconsider r = a`2 as Real;
    let s be Real;
    assume 0 < s;
    then
A6: a`2 - s < r - 0 by XREAL_1:15;
    take r;
    a in south_halfline a & r = proj2.a by PSCOMP_1:def 6,TOPREAL1:38;
    hence r in X by FUNCT_2:35;
    thus a`2 - s < r by A6;
  end;
  X is bounded_above by Th4;
  hence thesis by A1,A5,SEQ_4:def 1;
end;
