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

theorem
X c= dom f & f|X is continuous
  implies (||.f.||)|X is continuous & (-f)|X is continuous
proof
   assume that
A1: X c= dom f and
A2: f|X is continuous;
   thus (||.f.||)|X is continuous
   proof
    let r be Real;
    assume A3: r in dom((||.f.||)|X); then
A4:r in dom ||.f.|| & r in X by RELAT_1:57; then
    r in dom f by NORMSP_0:def 3; then
A5: r in dom(f|X) by A4,RELAT_1:57; then
A6: f|X is_continuous_in r by A2;
    thus (||.f.||)|X is_continuous_in r
    proof
     let s1;
     assume that
A7:   rng s1 c= dom ((||.f.||)|X) and
A8:   s1 is convergent & lim s1 = r;
     rng s1 c= dom (||.f.||) /\ X by A7,RELAT_1:61; then
     rng s1 c= dom f /\ X by NORMSP_0:def 3; then
A9:  rng s1 c= dom (f|X) by RELAT_1:61;
     now let n be Element of NAT;
A10:  s1.n in rng s1 by VALUED_0:28; then
      s1.n in dom (f|X) by A9; then
      s1.n in dom f /\ X by RELAT_1:61; then
A11:  s1.n in X & s1.n in dom f by XBOOLE_0:def 4; then
A12:   s1.n in dom (||.f.||) by NORMSP_0:def 3;
      thus ||. (f|X)/*s1 .|| .n = ||. ((f|X)/*s1).n .|| by NORMSP_0:def 4
          .=||.(f|X)/.(s1.n) .|| by A9,FUNCT_2:109
          .=||. f/.(s1.n) .|| by A9,A10,PARTFUN2:15
          .=(||.f.||).(s1.n) by A12,NORMSP_0:def 3
          .=((||.f.||)|X).(s1.n) by A11,FUNCT_1:49
          .=(((||.f.||)|X)/*s1).n by A7,FUNCT_2:108;
      end; then
A13:  ||.(f|X)/*s1 .|| = ((||.f.||)|X)/*s1 by FUNCT_2:63;
      r in REAL by XREAL_0:def 1;
      then
A14:  ||.(f|X)/.r.|| = ||. f/.r .|| by A5,PARTFUN2:15
        .= (||.f.||).r by A4,NORMSP_0:def 3
        .= ((||.f.||)|X).r by A3,FUNCT_1:47;
A15:  (f|X)/*s1 is convergent by A6,A8,A9;
      hence ((||.f.||)|X)/*s1 is convergent by A13,NORMSP_1:23;
      (f|X)/.r = lim ((f|X)/*s1) by A6,A8,A9;
      hence thesis by A15,A13,A14,LOPBAN_1:20;
     end;
    end;
    ((-1)(#)f)|X is continuous by A1,A2,Th21;
    hence thesis by VFUNCT_1:23;
end;
