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 that
A1: f | X is uniformly_continuous and
A2: X1 c= X ; :: thesis:
now :: thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X1) & x2 in dom (f | X1) & |.(x1 - x2).| < s holds
|.((f . x1) - (f . x2)).| < r ) )
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) & |.(x1 - x2).| < s holds
|.((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) & |.(x1 - x2).| < s holds
|.((f . x1) - (f . x2)).| < r ) )

then consider s being Real such that
A3: 0 < s and
A4: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds
|.((f . x1) - (f . x2)).| < r by ;
take s = s; :: thesis: ( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X1) & x2 in dom (f | X1) & |.(x1 - x2).| < s holds
|.((f . x1) - (f . x2)).| < r ) )

thus 0 < s by A3; :: thesis: for x1, x2 being Real st x1 in dom (f | X1) & x2 in dom (f | X1) & |.(x1 - x2).| < s holds
|.((f . x1) - (f . x2)).| < r

let x1, x2 be Real; :: thesis: ( x1 in dom (f | X1) & x2 in dom (f | X1) & |.(x1 - x2).| < s implies |.((f . x1) - (f . x2)).| < r )
assume that
A5: x1 in dom (f | X1) and
A6: x2 in dom (f | X1) and
A7: |.(x1 - x2).| < s ; :: thesis: |.((f . x1) - (f . x2)).| < r
f | X1 c= f | X by ;
then dom (f | X1) c= dom (f | X) by RELAT_1:11;
hence |.((f . x1) - (f . x2)).| < r by A4, A5, A6, A7; :: thesis: verum
end;
hence f | X1 is uniformly_continuous by Th1; :: thesis: verum