
theorem Th8:
  for X being non empty TopSpace,
  f1 being Function of X,R^1,a being Real 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=a-r1) & g is continuous
proof
  let X be non empty TopSpace, f1 be Function of X,R^1,a be Real;
  assume f1 is continuous;
  then consider g1 being Function of X,R^1 such that
A1: for p being Point of X,r1 being Real st f1.p=r1 holds g1.p=r1-a and
A2: g1 is continuous by Th7;
  consider g2 being Function of X,R^1 such that
A3: for p being Point of X,r1 being Real st g1.p=r1 holds g2.p= -r1 and
A4: g2 is continuous by A2,JGRAPH_4:8;
  for p being Point of X,r1 being Real st f1.p=r1 holds g2.p=a-r1
  proof
    let p be Point of X,r1 be Real;
    assume f1.p=r1;
    then g1.p=r1-a by A1;
    then g2.p=-(r1-a) by A3
      .=a-r1;
    hence thesis;
  end;
  hence thesis by A4;
end;
