let X, X1 be set ; :: thesis: for f being PartFunc of REAL ,REAL st f | X is uniformly_continuous & X1 c= X holds
f | X1 is uniformly_continuous
let f be PartFunc of REAL ,REAL ; :: thesis: ( f | X is uniformly_continuous & X1 c= X implies f | X1 is uniformly_continuous )
assume A1: ( f | X is uniformly_continuous & X1 c= X ) ; :: thesis: f | X1 is uniformly_continuous

now

let r be Real; :: thesis: ( 0 < r implies ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X1) & x2 in dom (f | X1) & abs (x1 - x2) < s holds
abs ((f . x1) - (f . x2)) < r ) ) )
assume 0 < r ; :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X1) & x2 in dom (f | X1) & abs (x1 - x2) < s holds
abs ((f . x1) - (f . x2)) < r ) )
then consider s being Real such that
A2: ( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & abs (x1 - x2) < s holds
abs ((f . x1) - (f . x2)) < r ) ) by A1, Def1;
f | X1 c= f | X by A1, RELAT_1:104;
then X: dom (f | X1) c= dom (f | X) by RELAT_1:25;
take s = s; :: thesis: ( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X1) & x2 in dom (f | X1) & abs (x1 - x2) < s holds
abs ((f . x1) - (f . x2)) < r ) )
thus 0 < s by A2; :: thesis: for x1, x2 being Real st x1 in dom (f | X1) & x2 in dom (f | X1) & abs (x1 - x2) < s holds
abs ((f . x1) - (f . x2)) < r
let x1, x2 be Real; :: thesis: ( x1 in dom (f | X1) & x2 in dom (f | X1) & abs (x1 - x2) < s implies abs ((f . x1) - (f . x2)) < r )
assume ( x1 in dom (f | X1) & x2 in dom (f | X1) & abs (x1 - x2) < s ) ; :: thesis: abs ((f . x1) - (f . x2)) < r
hence abs ((f . x1) - (f . x2)) < r by A2, X; :: thesis: verum

end;

hence f | X1 is uniformly_continuous by Def1; :: thesis: verum