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

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

let x1, x2 be Real; :: thesis: ( x1 in dom (f | X) & x2 in dom (f | X) implies |.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| )
assume ( x1 in dom (f | X) & x2 in dom (f | X) ) ; :: thesis: |.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).|
then ( x1 in dom f & x2 in dom f ) by RELAT_1:57;
hence |.((f . x1) - (f . x2)).| <= r * |.(x1 - x2).| by A2; :: thesis: verum

end;

hence for b₁ being PartFunc of REAL,REAL st b₁ = f | X holds
b₁ is Lipschitzian by A1, Th32; :: thesis: verum