reconsider g = f as PartFunc of REAL, the carrier of (REAL-NS n) by REAL_NS1:def 4;
g is Lipschitzian by Def6Th1;
then g | X is Lipschitzian ;
hence for b₁ being PartFunc of REAL,(REAL n) st b₁ = f | X holds
b₁ is Lipschitzian by Def6Th1; :: thesis: verum