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

let x1, x2 be Real; :: thesis: ( x1 in dom ||.f.|| & x2 in dom ||.f.|| implies |.((||.f.|| . x1) - (||.f.|| . x2)).| <= s * |.(x1 - x2).| )
assume A3: ( x1 in dom ||.f.|| & x2 in dom ||.f.|| ) ; :: thesis: |.((||.f.|| . x1) - (||.f.|| . x2)).| <= s * |.(x1 - x2).|
then ( x1 in dom f & x2 in dom f ) by NORMSP_0:def 3;
then A4: ||.((f /. x1) - (f /. x2)).|| <= s * |.(x1 - x2).| by A2;
|.((||.f.|| . x1) - (||.f.|| . x2)).| = |.(||.(f /. x1).|| - (||.f.|| . x2)).| by A3, NORMSP_0:def 3
.= |.(||.(f /. x1).|| - ||.(f /. x2).||).| by A3, NORMSP_0:def 3 ;
then |.((||.f.|| . x1) - (||.f.|| . x2)).| <= ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:9;
hence |.((||.f.|| . x1) - (||.f.|| . x2)).| <= s * |.(x1 - x2).| by A4, XXREAL_0:2; :: thesis: verum

end;

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