consider s being real number such that
A1: 0 < s and
A2: for x1, x2 being real number st x1 in dom f & x2 in dom f holds
abs ((f . x1) - (f . x2)) <= s * (abs (x1 - x2)) by Def3;

now

take s = s; :: thesis: ( 0 < s & ( for x1, x2 being real number st x1 in dom (abs f) & x2 in dom (abs f) holds
abs (((abs f) . x1) - ((abs f) . x2)) <= s * (abs (x1 - x2)) ) )
thus 0 < s by A1; :: thesis: for x1, x2 being real number st x1 in dom (abs f) & x2 in dom (abs f) holds
abs (((abs f) . x1) - ((abs f) . x2)) <= s * (abs (x1 - x2))
let x1, x2 be real number ; :: thesis: ( x1 in dom (abs f) & x2 in dom (abs f) implies abs (((abs f) . x1) - ((abs f) . x2)) <= s * (abs (x1 - x2)) )
assume ( x1 in dom (abs f) & x2 in dom (abs f) ) ; :: thesis: abs (((abs f) . x1) - ((abs f) . x2)) <= s * (abs (x1 - x2))
then ( x1 in dom f & x2 in dom f ) by VALUED_1:def 11;
then A3: abs ((f . x1) - (f . x2)) <= s * (abs (x1 - x2)) by A2;
abs (((abs f) . x1) - ((abs f) . x2)) = abs ((abs (f . x1)) - ((abs f) . x2)) by VALUED_1:18
.= abs ((abs (f . x1)) - (abs (f . x2))) by VALUED_1:18 ;
then abs (((abs f) . x1) - ((abs f) . x2)) <= abs ((f . x1) - (f . x2)) by COMPLEX1:64;
hence abs (((abs f) . x1) - ((abs f) . x2)) <= s * (abs (x1 - x2)) by A3, XXREAL_0:2; :: thesis: verum

end;

hence for b₁ being PartFunc of REAL,REAL st b₁ = abs f holds
b₁ is Lipschitzian by Def3; :: thesis: verum