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

now :: thesis: for x1, x2 being Real st x1 in dom f & x2 in dom f holds
||.((f /. x1) - (f /. x2)).|| <= 1 * |.(x1 - x2).|

let x1, x2 be Real; :: thesis: ( x1 in dom f & x2 in dom f implies ||.((f /. x1) - (f /. x2)).|| <= 1 * |.(x1 - x2).| )
assume A2: ( x1 in dom f & x2 in dom f ) ; :: thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * |.(x1 - x2).|
then f /. x1 = f . x1 by PARTFUN1:def 6
.= f . x2 by A1, A2, FUNCT_1:def 10
.= f /. x2 by A2, PARTFUN1:def 6 ;
then ||.((f /. x1) - (f /. x2)).|| = 0 by NORMSP_1:6;
hence ||.((f /. x1) - (f /. x2)).|| <= 1 * |.(x1 - x2).| by COMPLEX1:46; :: thesis: verum

end;

hence f is Lipschitzian ; :: thesis: verum