reserve C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  p for Point of TOP-REAL 2,
  n for Element of NAT;

theorem
  south_halfline p \ {p} is convex
proof
  set P = south_halfline p \ {p};
  P = {|[ p`1,r ]| where r is Real: r < p`2 }
  proof
    hereby
      let x be object;
      assume
A1:   x in P;
      then reconsider y = x as Point of TOP-REAL 2;
A2:   x in south_halfline p by A1,XBOOLE_0:def 5;
      then
A3:   y`1 = p`1 by TOPREAL1:def 12;
      then
A4:   x = |[ p`1,y`2 ]| by EUCLID:53;
A5:   not x in {p} by A1,XBOOLE_0:def 5;
A6:   now
        assume y`2 = p`2;
        then x = p by A3,TOPREAL3:6;
        hence contradiction by A5,TARSKI:def 1;
      end;
      y`2 <= p`2 by A2,TOPREAL1:def 12;
      then y`2 < p`2 by A6,XXREAL_0:1;
      hence x in {|[ p`1,r ]| where r is Real : r < p`2} by A4;
    end;
    let x be object;
    assume x in {|[ p`1,r ]| where r is Real : r < p`2 };
    then consider r being Real such that
A7: x = |[ p`1,r ]| and
A8: r < p`2;
    reconsider y = x as Point of TOP-REAL 2 by A7;
A9: y`2 = r by A7,EUCLID:52;
    then
A10: not x in {p} by A8,TARSKI:def 1;
    y`1 = p`1 by A7,EUCLID:52;
    then x in south_halfline p by A8,A9,TOPREAL1:def 12;
    hence thesis by A10,XBOOLE_0:def 5;
  end;
  hence thesis by Th5;
end;
