let X be set ; :: thesis: for f being PartFunc of REAL,REAL st X c= dom f & f | X is uniformly_continuous holds
f | X is continuous
let f be PartFunc of REAL,REAL; :: thesis: ( X c= dom f & f | X is uniformly_continuous implies f | X is continuous )
assume A1: X c= dom f ; :: thesis: ( not f | X is uniformly_continuous or f | X is continuous )
assume A2: f | X is uniformly_continuous ; :: thesis: f | X is continuous

now

let x0, r be real number ; :: thesis: ( x0 in X & 0 < r implies ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
abs ((f . x1) - (f . x0)) < r ) ) )
assume that
A3: x0 in X and
A4: 0 < r ; :: thesis: ex s being real number st
( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
abs ((f . x1) - (f . x0)) < r ) )
A5: x0 in dom (f | X) by A1, A3, RELAT_1:62;
r is Real by XREAL_0:def 1;
then consider s being Real such that
A6: 0 < s and
A7: 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 A2, A4, Th1;
reconsider s = s as real number ;
take s = s; :: thesis: ( 0 < s & ( for x1 being real number st x1 in X & abs (x1 - x0) < s holds
abs ((f . x1) - (f . x0)) < r ) )
thus 0 < s by A6; :: thesis: for x1 being real number st x1 in X & abs (x1 - x0) < s holds
abs ((f . x1) - (f . x0)) < r
let x1 be real number ; :: thesis: ( x1 in X & abs (x1 - x0) < s implies abs ((f . x1) - (f . x0)) < r )
assume that
A8: x1 in X and
A9: abs (x1 - x0) < s ; :: thesis: abs ((f . x1) - (f . x0)) < r
x1 in dom (f | X) by A1, A8, RELAT_1:62;
hence abs ((f . x1) - (f . x0)) < r by A7, A9, A5; :: thesis: verum

end;

hence f | X is continuous by A1, FCONT_1:14; :: thesis: verum