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 )
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 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 ;
(f | X) . x1 = f . x1 by ;
hence ( not |.(x1 - x2).| < s or |.((f . x1) - (f . x2)).| < r ) by A3, A4, A5, A6; :: thesis: verum
end;
assume A7: 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:
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 ;
(f | X) . x1 = f . x1 by ;
hence ( not |.(x1 - x2).| < s or |.(((f | X) . x1) - ((f | X) . x2)).| < r ) by A9, A10, A11, A12; :: thesis: verum