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

theorem Th39:
  ex f being Function of I[01],Closed-Interval-TSpace(-1,1) st f
is being_homeomorphism & (for r being Real st r in [.0,1.] holds f.r=2*r-1) & f
  .0=-1 & f.1=1
proof
  consider f being Function of I[01], Closed-Interval-TSpace(2*0+-1,2*1+-1)
  such that
A1: f is being_homeomorphism and
A2: for r being Real st r in [.0,1.] holds f.r=2*r+-1
     by Th36,TOPMETR:20;
A3: for r being Real st r in [.0,1.] holds f.r=2*r-1
  proof
    let r be Real;
    assume r in [.0,1.];
    hence f.r=2*r+-1 by A2
      .=2*r-1;
  end;
  1 in [.0,1.] by XXREAL_1:1;
  then
A4: f.1=2*1-1 by A3
    .=1;
  f.0=2*0-1 by A3,Lm1
    .=-1;
  hence thesis by A1,A3,A4;
end;
