let f be PartFunc of REAL ,REAL ; :: thesis: ( f is constant implies f is Lipschitzian )
assume A1: f is constant ; :: thesis: f is Lipschitzian

now

let x1, x2 be real number ; :: thesis: ( x1 in dom f & x2 in dom f implies abs ((f . x1) - (f . x2)) <= 1 * (abs (x1 - x2)) )
assume ( x1 in dom f & x2 in dom f ) ; :: thesis: abs ((f . x1) - (f . x2)) <= 1 * (abs (x1 - x2))
then f . x1 = f . x2 by A1, FUNCT_1:def 16;
then abs ((f . x1) - (f . x2)) = 0 by ABSVALUE:7;
hence abs ((f . x1) - (f . x2)) <= 1 * (abs (x1 - x2)) by COMPLEX1:132; :: thesis: verum

end;

hence f is Lipschitzian by Def3; :: thesis: verum