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

now :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom (abs f) & x2 in dom (abs f) holds
|.(((abs f) . x1) - ((abs f) . x2)).| <= s * |.(x1 - x2).| ) )

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