let X be set ; :: thesis: for Y being RealNormSpace
for f being PartFunc of REAL, the carrier of Y 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 Y be RealNormSpace; :: thesis: for f being PartFunc of REAL, the carrier of Y 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, the carrier of Y; :: 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 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 A4: ( x1 in dom (f | X) & x2 in dom (f | X) ) ; :: thesis: ( not |.(x1 - x2).| < s or ||.((f /. x1) - (f /. x2)).|| < r )
then ( (f | X) /. x1 = f /. x1 & (f | X) /. x2 = f /. x2 ) by PARTFUN1:80;
hence ( not |.(x1 - x2).| < s or ||.((f /. x1) - (f /. x2)).|| < r ) by A3, A4; :: thesis: verum

end;

assume A5: 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: f | X is uniformly_continuous
let r be Real; :: according to INTEGR20: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
A6: 0 < s and
A7: 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 A5;
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 A6; :: 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 A8: ( x1 in dom (f | X) & x2 in dom (f | X) ) ; :: thesis: ( not |.(x1 - x2).| < s or ||.(((f | X) /. x1) - ((f | X) /. x2)).|| < r )
then ( (f | X) /. x1 = f /. x1 & (f | X) /. x2 = f /. x2 ) by PARTFUN1:80;
hence ( not |.(x1 - x2).| < s or ||.(((f | X) /. x1) - ((f | X) /. x2)).|| < r ) by A7, A8; :: thesis: verum