theorem Th107:
  south_halfline p is non bounded
proof
  set Wp = south_halfline p;
  set p11 = p`1, p12 = p`2;
  assume Wp is bounded;
  then reconsider C = Wp as bounded Subset of Euclid 2 by Th5;
  consider r being Real such that
A1: 0 < r and
A2: for x,y being Point of Euclid 2 st x in C & y in C holds dist(x,y)
  <= r by TBSP_1:def 7;
  set EX2 = p`2-2*r, EX1 = p`1;
  reconsider p1 = p, EX = |[p`1, p`2-2*r]| as Point of Euclid 2 by EUCLID:67;
  p = |[p11,p12]| by EUCLID:53;
  then
A3: dist (p1, EX) = sqrt ((p11 - EX1)^2 + (p12 - EX2)^2) by GOBOARD6:6
    .= 2*r by A1,SQUARE_1:22;
A4: |[p`1, EX2]|`1 = p`1;
  then
A5: p1 in Wp by TOPREAL1:def 12;
  0 + p`2 <= 2*r + p`2 by A1,XREAL_1:6;
  then p`2 - 2*r <= p`2 by XREAL_1:20;
  then |[p`1, EX2]|`2 <= p`2;
  then EX in Wp by A4,TOPREAL1:def 12;
  hence thesis by A1,A2,A5,A3,XREAL_1:155;
end;
