reserve r,lambda for Real,
  i,j,n for Nat;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL 2,
  P, P1 for Subset of TOP-REAL 2;
reserve T for TopSpace;
reserve f,f1,f2,h for FinSequence of TOP-REAL 2;

theorem
  south_halfline p = {q where q is Point of TOP-REAL 2: q`1 = p`1 & q`2 <= p`2}
proof
  set A = {q where q is Point of TOP-REAL 2: q`1 = p`1 & q`2 <= p`2};
  hereby
    let x be object;
    assume
A1: x in south_halfline p;
    then reconsider q = x as Point of TOP-REAL 2;
A2: q`2 <= p`2 by A1,Def12;
    q`1 = p`1 by A1,Def12;
    hence x in A by A2;
  end;
  let x be object;
  assume x in A;
  then ex q being Point of TOP-REAL 2 st x = q & q`1 = p`1 & q`2 <= p`2;
  hence thesis by Def12;
end;
