theorem Th4:
  proj2.:(south_halfline a) is bounded_above
proof
  take a`2;
  let r be ExtReal;
  assume r in proj2.:(south_halfline a);
  then consider x being object such that
A1: x in the carrier of TOP-REAL 2 and
A2: x in south_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 12;
end;
