reserve n,m for Nat;
reserve x,X,X1 for set;
reserve s,g,r,p for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1,s2 for sequence of S;
reserve x0,x1,x2 for Point of S;
reserve Y for Subset of S;

theorem
  for f,Y st Y<>{} & Y c= dom f & Y is compact & f is_continuous_on Y ex
x1,x2 st x1 in Y & x2 in Y & ||.f.||/.x1 = upper_bound (||.f.||.:Y) & ||.f.||/.
  x2 = lower_bound (||.f.||.:Y)
proof
  let f,Y such that
A1: Y <> {} and
A2: Y c= dom f and
A3: Y is compact and
A4: f is_continuous_on Y;
A5: dom (f|Y) = dom f /\ Y by RELAT_1:61
    .= Y by A2,XBOOLE_1:28;
  f|Y is_continuous_on Y
  proof
    thus Y c= dom (f|Y) by A5;
    let r be Point of S;
    assume r in Y;
    then f|Y is_continuous_in r by A4;
    hence thesis by RELAT_1:72;
  end;
  then consider x1,x2 such that
A6: x1 in dom (f|Y) and
A7: x2 in dom (f|Y) and
A8: ||.(f|Y).||/.x1 = upper_bound (rng ||.(f|Y).||) & ||.(f|Y).||/.x2 =
  lower_bound (rng ||.(f|Y).||) by A1,A3,A5,Th34;
A9: dom f = dom ||.f.|| by NORMSP_0:def 3;
  take x1,x2;
  thus x1 in Y & x2 in Y by A5,A6,A7;
A10: ||.f.||.:Y =rng ( ||.f.|| | Y) by RELAT_1:115
    .=rng( ||.(f|Y).|| ) by Th35;
A11: x2 in dom ||.(f|Y).|| by A7,NORMSP_0:def 3;
  then
A12: ||.(f|Y).||/.x2 =||.(f|Y).||.x2 by PARTFUN1:def 6
    .=||. (f|Y)/.x2.|| by A11,NORMSP_0:def 3
    .=||. f/.x2.|| by A7,PARTFUN2:15
    .=||. f.||.x2 by A2,A5,A7,A9,NORMSP_0:def 3
    .=||. f.||/.x2 by A2,A5,A7,A9,PARTFUN1:def 6;
A13: x1 in dom ||.(f|Y).|| by A6,NORMSP_0:def 3;
  then ||.(f|Y).||/.x1 =||.(f|Y).||.x1 by PARTFUN1:def 6
    .=||. (f|Y)/.x1.|| by A13,NORMSP_0:def 3
    .=||. f/.x1.|| by A6,PARTFUN2:15
    .=||. f.||.x1 by A2,A5,A6,A9,NORMSP_0:def 3
    .=||. f.||/.x1 by A2,A5,A6,A9,PARTFUN1:def 6;
  hence thesis by A8,A12,A10;
end;
