reserve a for Real;
reserve p,q for Point of TOP-REAL 2;

theorem Th8:
  for X being non empty TopSpace, f1 being Function of X,R^1 st f1
is continuous holds ex g being Function of X,R^1 st (for p being Point of X,r1
  being Real st f1.p=r1 holds g.p=-r1) & g is continuous
proof
  let X be non empty TopSpace, f1 be Function of X,R^1;
  consider g1 being Function of X,R^1 such that
A1: for p being Point of X holds g1.p=0 and
A2: g1 is continuous by JGRAPH_2:20;
  assume f1 is continuous;
  then consider g2 being Function of X,R^1 such that
A3: for p being Point of X,r1,r2 being Real st g1.p=r1 & f1.p=r2
  holds g2.p=r1-r2 and
A4: g2 is continuous by A2,JGRAPH_2:21;
  for p being Point of X,r1 being Real st f1.p=r1 holds g2.p=-r1
  proof
    let p be Point of X,r1 be Real;
    assume
A5: f1.p=r1;
    g1.p=0 by A1;
    then g2.p=0-r1 by A3,A5;
    hence thesis;
  end;
  hence thesis by A4;
end;
