let T, S be RealNormSpace; :: thesis: for f being PartFunc of S,T
for Y being Subset of S st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of S st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
let f be PartFunc of S,T; :: thesis: for Y being Subset of S st Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y holds
ex x1, x2 being Point of S st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
let Y be Subset of S; :: thesis: ( Y <> {} & Y c= dom f & Y is compact & f is_continuous_on Y implies ex x1, x2 being Point of S st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) ) )
assume that
A1: Y <> {} and
A2: Y c= dom f and
A3: Y is compact and
A4: f is_continuous_on Y ; :: thesis: ex x1, x2 being Point of S st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
A5: dom (f | Y) = (dom f) /\ Y by RELAT_1:90
.= Y by A2, XBOOLE_1:28 ;
f | Y is_continuous_on Y

proof

thus Y c= dom (f | Y) by A5; :: according to NFCONT_1:def 11 :: thesis: for x0 being Point of S st x0 in Y holds
(f | Y) | Y is_continuous_in x0
let r be Point of S; :: thesis: ( r in Y implies (f | Y) | Y is_continuous_in r )
assume r in Y ; :: thesis: (f | Y) | Y is_continuous_in r
then f | Y is_continuous_in r by A4, Def11;
hence (f | Y) | Y is_continuous_in r by RELAT_1:101; :: thesis: verum

end;

then consider x1, x2 being Point of S such that
A6: ( x1 in dom (f | Y) & x2 in dom (f | Y) & ||.(f | Y).|| /. x1 = upper_bound (rng ||.(f | Y).||) & ||.(f | Y).|| /. x2 = lower_bound (rng ||.(f | Y).||) ) by A1, A3, A5, Th41;
A7: dom f = dom ||.f.|| by Def2;
take x1 ; :: thesis: ex x2 being Point of S st
( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
take x2 ; :: thesis: ( x1 in Y & x2 in Y & ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
thus ( x1 in Y & x2 in Y ) by A5, A6; :: thesis: ( ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) )
A8: ( x1 in dom ||.(f | Y).|| & x2 in dom ||.(f | Y).|| ) by A6, Def2;
then A9: ||.(f | Y).|| /. x1 = ||.(f | Y).|| . x1 by PARTFUN1:def 8
.= ||.((f | Y) /. x1).|| by A8, Def2
.= ||.(f /. x1).|| by A6, PARTFUN2:32
.= ||.f.|| . x1 by A2, A5, A6, A7, Def2
.= ||.f.|| /. x1 by A2, A5, A6, A7, PARTFUN1:def 8 ;
A10: ||.(f | Y).|| /. x2 = ||.(f | Y).|| . x2 by A8, PARTFUN1:def 8
.= ||.((f | Y) /. x2).|| by A8, Def2
.= ||.(f /. x2).|| by A6, PARTFUN2:32
.= ||.f.|| . x2 by A2, A5, A6, A7, Def2
.= ||.f.|| /. x2 by A2, A5, A6, A7, PARTFUN1:def 8 ;
||.f.|| .: Y = rng (||.f.|| | Y) by RELAT_1:148
.= rng ||.(f | Y).|| by Th42 ;
hence ( ||.f.|| /. x1 = upper_bound (||.f.|| .: Y) & ||.f.|| /. x2 = lower_bound (||.f.|| .: Y) ) by A6, A9, A10; :: thesis: verum