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
  east_halfline p \ {p} is convex
proof
  set P = east_halfline p \ {p};
  P = {|[ r,p`2 ]| where r is Real : r > p`1 }
  proof
    hereby
      let x be object;
      assume
A1:   x in P;
      then reconsider y = x as Point of TOP-REAL 2;
A2:   x in east_halfline p by A1,XBOOLE_0:def 5;
      then
A3:   y`2 = p`2 by TOPREAL1:def 11;
      then
A4:   x = |[ y`1,p`2 ]| by EUCLID:53;
A5:   not x in {p} by A1,XBOOLE_0:def 5;
A6:   now
        assume y`1 = p`1;
        then x = p by A3,TOPREAL3:6;
        hence contradiction by A5,TARSKI:def 1;
      end;
      y`1 >= p`1 by A2,TOPREAL1:def 11;
      then y`1 > p`1 by A6,XXREAL_0:1;
      hence x in {|[ r,p`2 ]| where r is Real : r > p`1} by A4;
    end;
    let x be object;
    assume x in {|[ r,p`2 ]| where r is Real : r > p`1 };
    then consider r being Real such that
A7: x = |[ r,p`2 ]| and
A8: r > p`1;
    reconsider y = x as Point of TOP-REAL 2 by A7;
A9: y`1 = r by A7,EUCLID:52;
    then
A10: not x in {p} by A8,TARSKI:def 1;
    y`2 = p`2 by A7,EUCLID:52;
    then x in east_halfline p by A8,A9,TOPREAL1:def 11;
    hence thesis by A10,XBOOLE_0:def 5;
  end;
  hence thesis by Th2;
end;
