let X, X1 be set ; :: thesis: for f being PartFunc of REAL,REAL st f | X is Lipschitzian & X1 c= X holds

f | X1 is Lipschitzian

let f be PartFunc of REAL,REAL; :: thesis: ( f | X is Lipschitzian & X1 c= X implies f | X1 is Lipschitzian )

assume that

A1: f | X is Lipschitzian and

A2: X1 c= X ; :: thesis: f | X1 is Lipschitzian

f | X1 = (f | X) | X1 by A2, RELAT_1:74;

hence f | X1 is Lipschitzian by A1; :: thesis: verum

f | X1 is Lipschitzian

let f be PartFunc of REAL,REAL; :: thesis: ( f | X is Lipschitzian & X1 c= X implies f | X1 is Lipschitzian )

assume that

A1: f | X is Lipschitzian and

A2: X1 c= X ; :: thesis: f | X1 is Lipschitzian

f | X1 = (f | X) | X1 by A2, RELAT_1:74;

hence f | X1 is Lipschitzian by A1; :: thesis: verum