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 Th35:
  south_halfline p = { |[ p`1,r ]| where r is Real: r <= p`2 }
proof
  set A = {|[ p`1,r ]| where r is Real: r <= 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;
A3: q = |[q`1, q`2]| by EUCLID:53;
    q`1 = p`1 by A1,Def12;
    hence x in A by A2,A3;
  end;
  let x be object;
  assume x in A;
  then consider r being Real such that
A4: x = |[p`1,r]| and
A5: r <= p`2;
  reconsider q = x as Point of TOP-REAL 2 by A4;
A6: q`2 = r by A4,EUCLID:52;
  q`1 = p`1 by A4,EUCLID:52;
  hence thesis by A5,A6,Def12;
end;
