
theorem Th53:
  for a,b,c,d being Real, h being Function of TOP-REAL 2,
TOP-REAL 2,f being Function of I[01],TOP-REAL 2 st a<b & c <d & h=AffineMap(2/(
b-a),-(b+a)/(b-a),2/(d-c),-(d+c)/(d-c)) & f is continuous one-to-one holds h*f
  is continuous one-to-one
proof
  let a,b,c,d be Real, h be Function of TOP-REAL 2,TOP-REAL 2,f be
  Function of I[01],TOP-REAL 2;
  set A=2/(b-a), B=-(b+a)/(b-a), C = 2/(d-c), D=-(d+c)/(d-c);
  assume that
A1: a<b and
A2: c <d and
A3: h=AffineMap(A,B,C,D) and
A4: f is continuous one-to-one;
  d-c >0 by A2,XREAL_1:50;
  then
A5: C >0 by XREAL_1:139;
  b-a>0 by A1,XREAL_1:50;
  then A >0 by XREAL_1:139;
  then h is being_homeomorphism by A3,A5,Th51;
  then h is one-to-one by TOPS_2:def 5;
  hence thesis by A3,A4,FUNCT_1:24;
end;
