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

now

let x1, x2 be real number ; :: thesis: ( x1 in dom (f | X) & x2 in dom (f | X) implies ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) )
assume ( x1 in dom (f | X) & x2 in dom (f | X) ) ; :: thesis: ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2))
then ( x1 in dom f & x2 in dom f ) by RELAT_1:86;
hence ||.((f /. x1) - (f /. x2)).|| <= r * (abs (x1 - x2)) by A2; :: thesis: verum

end;

hence for b₁ being PartFunc of REAL, the carrier of S st b₁ = f | X holds
b₁ is Lipschitzian by A1, Th33; :: thesis: verum