let X be set ; :: thesis: for f being PartFunc of REAL,REAL holds

( f | X is uniformly_continuous iff 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 | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.((f . x1) - (f . x2)).| < r ) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( f | X is uniformly_continuous iff 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 | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.((f . x1) - (f . x2)).| < r ) ) )

thus ( f | X is uniformly_continuous implies 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 | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.((f . x1) - (f . x2)).| < r ) ) ) :: 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 | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.((f . x1) - (f . x2)).| < r ) ) ) implies f | X is uniformly_continuous )

ex s being Real st

( 0 < s & ( 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 ) ) ; :: thesis: f | X is uniformly_continuous

let r be Real; :: according to FCONT_2:def 1 :: thesis: ( 0 < r implies ex s being Real st

( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.(((f | X) . x1) - ((f | X) . x2)).| < r ) ) )

assume 0 < r ; :: thesis: ex s being Real st

( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.(((f | X) . x1) - ((f | X) . x2)).| < r ) )

then consider s being Real such that

A8: 0 < s and

A9: 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 A7;

take s ; :: thesis: ( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.(((f | X) . x1) - ((f | X) . x2)).| < r ) )

thus 0 < s by A8; :: thesis: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.(((f | X) . x1) - ((f | X) . x2)).| < r

let x1, x2 be Real; :: thesis: ( x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s implies |.(((f | X) . x1) - ((f | X) . x2)).| < r )

assume that

A10: x1 in dom (f | X) and

A11: x2 in dom (f | X) ; :: thesis: ( not |.(x1 - x2).| < s or |.(((f | X) . x1) - ((f | X) . x2)).| < r )

A12: (f | X) . x2 = f . x2 by A11, FUNCT_1:47;

(f | X) . x1 = f . x1 by A10, FUNCT_1:47;

hence ( not |.(x1 - x2).| < s or |.(((f | X) . x1) - ((f | X) . x2)).| < r ) by A9, A10, A11, A12; :: thesis: verum

( f | X is uniformly_continuous iff 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 | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.((f . x1) - (f . x2)).| < r ) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( f | X is uniformly_continuous iff 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 | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.((f . x1) - (f . x2)).| < r ) ) )

thus ( f | X is uniformly_continuous implies 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 | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.((f . x1) - (f . x2)).| < r ) ) ) :: 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 | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.((f . x1) - (f . x2)).| < r ) ) ) implies f | X is uniformly_continuous )

proof

assume A7:
for r being Real st 0 < r holds
assume A1:
f | X is uniformly_continuous
; :: 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 | X) & x2 in dom (f | X) & |.(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 | X) & x2 in dom (f | X) & |.(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 | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.((f . x1) - (f . x2)).| < r ) )

then consider s being Real such that

A2: 0 < s and

A3: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.(((f | X) . x1) - ((f | X) . x2)).| < r by A1;

take s ; :: thesis: ( 0 < s & ( 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 ) )

thus 0 < s by A2; :: thesis: 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

let x1, x2 be Real; :: thesis: ( x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s implies |.((f . x1) - (f . x2)).| < r )

assume that

A4: x1 in dom (f | X) and

A5: x2 in dom (f | X) ; :: thesis: ( not |.(x1 - x2).| < s or |.((f . x1) - (f . x2)).| < r )

A6: (f | X) . x2 = f . x2 by A5, FUNCT_1:47;

(f | X) . x1 = f . x1 by A4, FUNCT_1:47;

hence ( not |.(x1 - x2).| < s or |.((f . x1) - (f . x2)).| < r ) by A3, A4, A5, A6; :: thesis: verum

end;ex s being Real st

( 0 < s & ( 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 ) )

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 | X) & x2 in dom (f | X) & |.(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 | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.((f . x1) - (f . x2)).| < r ) )

then consider s being Real such that

A2: 0 < s and

A3: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.(((f | X) . x1) - ((f | X) . x2)).| < r by A1;

take s ; :: thesis: ( 0 < s & ( 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 ) )

thus 0 < s by A2; :: thesis: 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

let x1, x2 be Real; :: thesis: ( x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s implies |.((f . x1) - (f . x2)).| < r )

assume that

A4: x1 in dom (f | X) and

A5: x2 in dom (f | X) ; :: thesis: ( not |.(x1 - x2).| < s or |.((f . x1) - (f . x2)).| < r )

A6: (f | X) . x2 = f . x2 by A5, FUNCT_1:47;

(f | X) . x1 = f . x1 by A4, FUNCT_1:47;

hence ( not |.(x1 - x2).| < s or |.((f . x1) - (f . x2)).| < r ) by A3, A4, A5, A6; :: thesis: verum

ex s being Real st

( 0 < s & ( 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 ) ) ; :: thesis: f | X is uniformly_continuous

let r be Real; :: according to FCONT_2:def 1 :: thesis: ( 0 < r implies ex s being Real st

( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.(((f | X) . x1) - ((f | X) . x2)).| < r ) ) )

assume 0 < r ; :: thesis: ex s being Real st

( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.(((f | X) . x1) - ((f | X) . x2)).| < r ) )

then consider s being Real such that

A8: 0 < s and

A9: 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 A7;

take s ; :: thesis: ( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.(((f | X) . x1) - ((f | X) . x2)).| < r ) )

thus 0 < s by A8; :: thesis: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s holds

|.(((f | X) . x1) - ((f | X) . x2)).| < r

let x1, x2 be Real; :: thesis: ( x1 in dom (f | X) & x2 in dom (f | X) & |.(x1 - x2).| < s implies |.(((f | X) . x1) - ((f | X) . x2)).| < r )

assume that

A10: x1 in dom (f | X) and

A11: x2 in dom (f | X) ; :: thesis: ( not |.(x1 - x2).| < s or |.(((f | X) . x1) - ((f | X) . x2)).| < r )

A12: (f | X) . x2 = f . x2 by A11, FUNCT_1:47;

(f | X) . x1 = f . x1 by A10, FUNCT_1:47;

hence ( not |.(x1 - x2).| < s or |.(((f | X) . x1) - ((f | X) . x2)).| < r ) by A9, A10, A11, A12; :: thesis: verum