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 Th21:
X c= dom f & f|X is continuous implies (r(#)f)|X is continuous
proof
   assume that
A1: X c= dom f and
A2: f|X is continuous;
A3:X c= dom(r(#)f) by A1,VFUNCT_1:def 4;
   now let s1;
    assume that
A4: rng s1 c= X & s1 is convergent and
A5: lim s1 in X;
A6: f/*s1 is convergent by A1,A2,A4,A5,Th16; then
A7: r*(f/*s1) is convergent by NORMSP_1:22;
A8:   lim s1 in REAL by XREAL_0:def 1;
    f/.(lim s1) = lim (f/*s1) by A1,A2,A4,A5,Th16;
    then (r(#)f)/.(lim s1) = r * lim (f/*s1) by A3,A5,VFUNCT_1:def 4,A8
       .= lim (r*(f/*s1)) by A6,NORMSP_1:28
       .= lim ((r(#)f)/*s1) by A1,A4,Th4,XBOOLE_1:1;
    hence (r(#)f)/*s1 is convergent & (r(#)f)/.(lim s1)=lim((r(#)f)/*s1)
      by A1,A4,A7,Th4,XBOOLE_1:1;
   end;
   hence thesis by A3,Th16;
end;
