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
||.((f /. x1) - (f /. x2)).|| <= s * (abs (x1 - x2)) by Def3;

now

let x1, x2 be real number ; :: thesis: ( x1 in dom ||.f.|| & x2 in dom ||.f.|| implies abs ((||.f.|| . x1) - (||.f.|| . x2)) <= s * (abs (x1 - x2)) )
assume A4: ( x1 in dom ||.f.|| & x2 in dom ||.f.|| ) ; :: thesis: abs ((||.f.|| . x1) - (||.f.|| . x2)) <= s * (abs (x1 - x2))
then ( x1 in dom f & x2 in dom f ) by NORMSP_0:def 3;
then A3: ||.((f /. x1) - (f /. x2)).|| <= s * (abs (x1 - x2)) by A2;
abs ((||.f.|| . x1) - (||.f.|| . x2)) = abs (||.(f /. x1).|| - (||.f.|| . x2)) by A4, NORMSP_0:def 3
.= abs (||.(f /. x1).|| - ||.(f /. x2).||) by A4, NORMSP_0:def 3 ;
then abs ((||.f.|| . x1) - (||.f.|| . x2)) <= ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:13;
hence abs ((||.f.|| . x1) - (||.f.|| . x2)) <= s * (abs (x1 - x2)) by A3, XXREAL_0:2; :: thesis: verum

end;

hence for b₁ being PartFunc of REAL,REAL st b₁ = ||.f.|| holds
b₁ is Lipschitzian by FCONT_1:def 3, A1; :: thesis: verum