reserve C for Simple_closed_curve,
  A,A1,A2 for Subset of TOP-REAL 2,
  p,p1,p2,q ,q1,q2 for Point of TOP-REAL 2,
  n for Element of NAT;

theorem Th20:
  for f being Function of R^1, R^1 for a,b being Real st a <> 0 &
  f = AffineMap(a,b) holds f is being_homeomorphism
proof
  let f be Function of R^1, R^1;
  let a,b be Real such that
A1: a <> 0 and
A2: f = AffineMap(a,b);
  thus dom f = [#]R^1 by FUNCT_2:def 1;
  thus
A3: rng f = [#]R^1 by A1,A2,FCONT_1:55,TOPMETR:17;
  thus
A4: f is one-to-one by A1,A2,FCONT_1:50;
  for x being Real holds f.x = a*x + b by A2,FCONT_1:def 4;
  hence f is continuous by TOPMETR:21;
  f is onto by A3,FUNCT_2:def 3;
  then f" = (f qua Function)" by A4,TOPS_2:def 4
    .= AffineMap(a",-b/a) by A1,A2,FCONT_1:56;
  then for x being Real holds f".x = a"*x + -b/a by FCONT_1:def 4;
  hence thesis by TOPMETR:21;
end;
