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

theorem Th38:
  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)*1+1,(-2)*0+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 Th37,TOPMETR:20;
  1 in [.0,1.] by XXREAL_1:1;
  then
A3: f.1=-1 by A2;
  f.0=(-2)*0+1 by A2,Lm1;
  hence thesis by A1,A2,A3;
end;
