reserve n,m,i,k for Element of NAT;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,x0,x1,x2 for Real;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve h for PartFunc of REAL,REAL-NS n;
reserve W for non empty set;

theorem Th22:
for f1 be PartFunc of REAL,REAL n,
    f2 be PartFunc of REAL n,REAL st x0 in dom (f2*f1) &
 f1 is_continuous_in x0 & f2 is_continuous_in f1/.x0
 holds f2*f1 is_continuous_in x0
proof
  let f1 be PartFunc of REAL,REAL n,
      f2 be PartFunc of REAL n,REAL;
  assume
A1: x0 in dom (f2*f1) & f1 is_continuous_in x0
     & f2 is_continuous_in f1/.x0;
  reconsider g1=f1 as PartFunc of REAL,REAL-NS n
    by REAL_NS1:def 4;
  reconsider g2=f2 as PartFunc of REAL-NS n,REAL
     by REAL_NS1:def 4;
    f1/.x0 = g1/.x0 by REAL_NS1:def 4;
   then g2 is_continuous_in g1/.x0 by A1;
   hence thesis by A1,Th20;
end;
