theorem Th6:
  proj1.:(east_halfline a) is bounded_below
proof
  take a`1;
  let r be ExtReal;
  assume r in proj1.:(east_halfline a);
  then consider x being object such that
A1: x in the carrier of TOP-REAL 2 and
A2: x in east_halfline a and
A3: r = proj1.x by FUNCT_2:64;
  reconsider x as Point of TOP-REAL 2 by A1;
  r = x`1 by A3,PSCOMP_1:def 5;
  hence thesis by A2,TOPREAL1:def 11;
end;
