let S be RealNormSpace; :: thesis: for f being PartFunc of the carrier of S,REAL st dom f <> {} & dom f is compact & f is_continuous_on dom f holds
ex x1, x2 being Point of S st
( x1 in dom f & x2 in dom f & f /. x1 = upper_bound (rng f) & f /. x2 = lower_bound (rng f) )
let f be PartFunc of the carrier of S,REAL; :: thesis: ( dom f <> {} & dom f is compact & f is_continuous_on dom f implies ex x1, x2 being Point of S st
( x1 in dom f & x2 in dom f & f /. x1 = upper_bound (rng f) & f /. x2 = lower_bound (rng f) ) )
assume ( dom f <> {} & dom f is compact & f is_continuous_on dom f ) ; :: thesis: ex x1, x2 being Point of S st
( x1 in dom f & x2 in dom f & f /. x1 = upper_bound (rng f) & f /. x2 = lower_bound (rng f) )
then A1: ( rng f <> {} & rng f is compact ) by Th31, RELAT_1:42;
then consider x being Element of S such that
A2: ( x in dom f & upper_bound (rng f) = f . x ) by PARTFUN1:3, RCOMP_1:14;
take x ; :: thesis: ex x2 being Point of S st
( x in dom f & x2 in dom f & f /. x = upper_bound (rng f) & f /. x2 = lower_bound (rng f) )
consider y being Element of S such that
A3: ( y in dom f & lower_bound (rng f) = f . y ) by A1, PARTFUN1:3, RCOMP_1:14;
take y ; :: thesis: ( x in dom f & y in dom f & f /. x = upper_bound (rng f) & f /. y = lower_bound (rng f) )
thus ( x in dom f & y in dom f & f /. x = upper_bound (rng f) & f /. y = lower_bound (rng f) ) by A2, A3, PARTFUN1:def 6; :: thesis: verum