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 Th13:
f is_continuous_in x0 implies r(#)f is_continuous_in x0
proof
   assume A1: f is_continuous_in x0; then
   x0 in dom f;
   hence A2: x0 in dom (r(#)f) by VFUNCT_1:def 4;
   let s1;
   assume that
A3: rng s1 c= dom(r(#)f) and
A4: s1 is convergent & lim s1=x0;
A5:rng s1 c= dom f by A3,VFUNCT_1:def 4; then
A6:f/.x0 = lim (f/*s1) by A1,A4;
A7:f/*s1 is convergent by A1,A4,A5; then
   r*(f/*s1) is convergent by NORMSP_1:22;
   hence (r(#)f)/*s1 is convergent by A5,Th4;
   thus (r(#)f)/.x0 = r*f/.x0 by A2,VFUNCT_1:def 4
    .= lim (r*(f/*s1)) by A7,A6,NORMSP_1:28
    .= lim ((r(#)f)/*s1) by A5,Th4;
end;
