let x, t be Real; :: thesis: ( 0 < t implies for f being PartFunc of REAL ,REAL st [.x,(x + t).] c= dom f & f | [.x,(x + t).] is continuous & f . x = f . (x + t) & f is_differentiable_on ].x,(x + t).[ holds
ex s being Real st
( 0 < s & s < 1 & diff f,(x + (s * t)) = 0 ) )
assume A1: 0 < t ; :: thesis: for f being PartFunc of REAL ,REAL st [.x,(x + t).] c= dom f & f | [.x,(x + t).] is continuous & f . x = f . (x + t) & f is_differentiable_on ].x,(x + t).[ holds
ex s being Real st
( 0 < s & s < 1 & diff f,(x + (s * t)) = 0 )
let f be PartFunc of REAL ,REAL ; :: thesis: ( [.x,(x + t).] c= dom f & f | [.x,(x + t).] is continuous & f . x = f . (x + t) & f is_differentiable_on ].x,(x + t).[ implies ex s being Real st
( 0 < s & s < 1 & diff f,(x + (s * t)) = 0 ) )
assume that
X: [.x,(x + t).] c= dom f and
A2: f | [.x,(x + t).] is continuous and
A3: f . x = f . (x + t) and
A4: f is_differentiable_on ].x,(x + t).[ ; :: thesis: ex s being Real st
( 0 < s & s < 1 & diff f,(x + (s * t)) = 0 )
consider x0 being Real such that
A5: ( x0 in ].x,(x + t).[ & diff f,x0 = 0 ) by A2, A3, A4, Th1, A1, X, XREAL_1:31;
take s = (x0 - x) / t; :: thesis: ( 0 < s & s < 1 & diff f,(x + (s * t)) = 0 )
x0 in { r where r is Real : ( x < r & r < x + t ) } by A5, RCOMP_1:def 2;
then A6: ex g being Real st
( g = x0 & x < g & g < x + t ) ;
then 0 < x0 - x by XREAL_1:52;
then 0 / t < (x0 - x) / t by A1, XREAL_1:76;
hence 0 < s ; :: thesis: ( s < 1 & diff f,(x + (s * t)) = 0 )
x0 - x < t by A6, XREAL_1:21;
then (x0 - x) / t < t / t by A1, XREAL_1:76;
hence s < 1 by A1, XCMPLX_1:60; :: thesis: diff f,(x + (s * t)) = 0
(s * t) + x = (x0 - x) + x by A1, XCMPLX_1:88;
hence diff f,(x + (s * t)) = 0 by A5; :: thesis: verum