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
x0 in dom f & f is_continuous_in x0
 implies ||.f.|| is_continuous_in x0 & -f is_continuous_in x0
proof
   assume A1: x0 in dom f;
   assume A2: f is_continuous_in x0;
A3:x0 in dom ||.f.|| by A1,NORMSP_0:def 3;
   now let s1;
    assume that
A4:  rng s1 c= dom ||.f.|| and
A5:  s1 is convergent & lim s1=x0;
A6: rng s1 c= dom f by A4,NORMSP_0:def 3; then
A7: f/.x0 = lim (f/*s1) by A2,A5;
A8: f/*s1 is convergent by A2,A5,A6; then
    ||.f/*s1.|| is convergent by NORMSP_1:23;
    hence ||.f.||/*s1 is convergent by A6,Th5;
    thus (||.f.||).x0 =||. f/.x0 .|| by A3,NORMSP_0:def 3
      .= lim ||.f/*s1.|| by A8,A7,LOPBAN_1:20
      .= lim (||.f.||/*s1) by A6,Th5;
   end;
   hence ||.f.|| is_continuous_in x0;
   (-1)(#)f is_continuous_in x0 by A2,Th13;
   hence thesis by VFUNCT_1:23;
end;
