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 Th3:
  proj2.:(north_halfline a) is bounded_below
proof
  take a`2;
  let r be ExtReal;
  assume r in proj2.:(north_halfline a);
  then consider x being object such that
A1: x in the carrier of TOP-REAL 2 and
A2: x in north_halfline a and
A3: r = proj2.x by FUNCT_2:64;
  reconsider x as Point of TOP-REAL 2 by A1;
  r = x`2 by A3,PSCOMP_1:def 6;
  hence thesis by A2,TOPREAL1:def 10;
end;
