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: f | X1 is uniformly_continuous

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: f | X1 is uniformly_continuous

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 ) )

hence
f | X1 is uniformly_continuous
by Th1; :: thesis: verumex 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 A1, Th1;

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 A2, RELAT_1:75;

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;( 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 A1, Th1;

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 A2, RELAT_1:75;

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