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
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 )
by A1, A2, A3; verum