reserve x for object;
reserve x0,r,r1,r2,g,g1,g2,p,y0 for Real;
reserve n,m,k,l for Element of NAT;
reserve a,b,d for Real_Sequence;
reserve h,h1,h2 for non-zero 0-convergent Real_Sequence;
reserve c,c1 for constant Real_Sequence;
reserve A for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve L for LinearFunc;
reserve R for RestFunc;

theorem Th9:
  rng a c= dom (f2*f1) implies rng a c= dom f1 & rng (f1/*a) c= dom f2
proof
  assume
A1: rng a c= dom (f2*f1);
  then
A2: f1.:(rng a) c= dom f2 by FUNCT_1:101;
  rng a c= dom f1 by A1,FUNCT_1:101;
  hence thesis by A2,VALUED_0:30;
end;
