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;

_{1} being PartFunc of REAL, the carrier of S st b_{1} = f | X holds

b_{1} is Lipschitzian
by A1, Th26; :: thesis: verum

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).|

hence
for b||.((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;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

b