reserve r, s, t, g for Real,

          r3, r1, r2, q3, p3 for Real;
reserve T for TopStruct,
  f for RealMap of T;

theorem Th14:
  for T being non empty TopSpace holds (for f being RealMap of T
  st f is continuous holds f is with_max) iff for f being RealMap of T st f is
  continuous holds f is with_min
proof
  let T be non empty TopSpace;
  hereby
    assume
A1: for f being RealMap of T st f is continuous holds f is with_max;
    let f be RealMap of T;
    assume f is continuous;
    then -f is continuous by Th9;
    then -f is with_max by A1;
    hence f is with_min by MEASURE6:66;
  end;
  assume
A2: for f being RealMap of T st f is continuous holds f is with_min;
  let f be RealMap of T;
  assume f is continuous;
  then -f is continuous by Th9;
  then - -f is with_max by A2,Lm2;
  hence thesis;
end;
