reserve n,m,k for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,p,x0,x1,x2 for Real;
reserve s1,s2,q1 for Real_Sequence;
reserve Y for Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th20:
  for r,X,f st X c= dom f & f|X is continuous holds (r(#)f)|X is continuous
proof
  let r,X,f such that
A1: X c= dom f;
  assume
A2: f|X is continuous;
A3: X c= dom(r(#)f) by A1,VALUED_1:def 5;
  now
    let s1;
    assume that
A4: rng s1 c= X and
A5: s1 is convergent and
A6: lim s1 in X;
A7: f/*s1 is convergent by A1,A2,A4,A5,A6,Th13;
    then
A8: r(#)(f/*s1) is convergent;
    f.(lim s1) = lim (f/*s1) by A1,A2,A4,A5,A6,Th13;
    then (r(#)f).(lim s1) = r * lim (f/*s1) by A3,A6,VALUED_1:def 5
      .= lim (r(#)(f/*s1)) by A7,SEQ_2:8
      .= lim ((r(#)f)/*s1) by A1,A4,RFUNCT_2:9,XBOOLE_1:1;
    hence (r(#)f)/*s1 is convergent & (r(#)f).(lim s1)=lim((r(#)f)/*s1) by A1
,A4,A8,RFUNCT_2:9,XBOOLE_1:1;
  end;
  hence thesis by A3,Th13;
end;
