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) & abs (x1 - x2) < s holds
abs ((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) & abs (x1 - x2) < s holds
abs ((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) & abs (x1 - x2) < s holds
abs ((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) & abs (x1 - x2) < s holds
abs ((f . x1) - (f . x2)) < r ) ) ) implies f | X is uniformly_continuous )

proof

assume Z1: 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) & abs (x1 - x2) < s holds
abs ((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) & abs (x1 - x2) < s holds
abs ((f . x1) - (f . x2)) < r ) ) )
assume Z2: 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) & abs (x1 - x2) < s holds
abs ((f . x1) - (f . x2)) < r ) )
consider s being Real such that
W1: 0 < s and
W2: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & abs (x1 - x2) < s holds
abs (((f | X) . x1) - ((f | X) . x2)) < r by Z1, Z2, Xdef;
take s ; :: thesis: ( 0 < s & ( 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 ) )
thus 0 < s by W1; :: thesis: 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
let x1, x2 be Real; :: thesis: ( x1 in dom (f | X) & x2 in dom (f | X) & abs (x1 - x2) < s implies abs ((f . x1) - (f . x2)) < r )
assume Z: ( x1 in dom (f | X) & x2 in dom (f | X) ) ; :: thesis: ( not abs (x1 - x2) < s or abs ((f . x1) - (f . x2)) < r )
then ( (f | X) . x1 = f . x1 & (f | X) . x2 = f . x2 ) by FUNCT_1:70;
hence ( not abs (x1 - x2) < s or abs ((f . x1) - (f . x2)) < r ) by W2, Z; :: thesis: verum

end;

assume Z0: 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) & abs (x1 - x2) < s holds
abs ((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) & abs (x1 - x2) < s holds
abs (((f | X) . x1) - ((f | X) . x2)) < r ) ) )
assume Z1: 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) & abs (x1 - x2) < s holds
abs (((f | X) . x1) - ((f | X) . x2)) < r ) )
consider s being Real such that
W1: 0 < s and
W2: 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 Z0, Z1;
take s ; :: thesis: ( 0 < s & ( for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & abs (x1 - x2) < s holds
abs (((f | X) . x1) - ((f | X) . x2)) < r ) )
thus 0 < s by W1; :: thesis: for x1, x2 being Real st x1 in dom (f | X) & x2 in dom (f | X) & abs (x1 - x2) < s holds
abs (((f | X) . x1) - ((f | X) . x2)) < r
let x1, x2 be Real; :: thesis: ( x1 in dom (f | X) & x2 in dom (f | X) & abs (x1 - x2) < s implies abs (((f | X) . x1) - ((f | X) . x2)) < r )
assume Z: ( x1 in dom (f | X) & x2 in dom (f | X) ) ; :: thesis: ( not abs (x1 - x2) < s or abs (((f | X) . x1) - ((f | X) . x2)) < r )
then ( (f | X) . x1 = f . x1 & (f | X) . x2 = f . x2 ) by FUNCT_1:70;
hence ( not abs (x1 - x2) < s or abs (((f | X) . x1) - ((f | X) . x2)) < r ) by Z, W2; :: thesis: verum