reserve r,r1,r2,g,g1,g2,x0 for Real;
reserve f1,f2 for PartFunc of REAL,REAL;

theorem Th1:
  for s be Real_Sequence,X be set st rng s c= dom(f2*f1) /\ X holds
rng s c= dom(f2*f1) & rng s c= X & rng s c= dom f1 & rng s c= dom f1 /\ X & rng
  (f1/*s) c= dom f2
proof
  let s be Real_Sequence,X be set;
  assume rng s c=dom(f2*f1)/\X;
  hence
A1: rng s c=dom(f2*f1) & rng s c=X by XBOOLE_1:18;
A2: dom(f2*f1)c=dom f1 by RELAT_1:25;
  hence rng s c=dom f1 by A1;
  hence rng s c=dom f1/\X by A1,XBOOLE_1:19;
  let x be object;
  assume x in rng(f1/*s);
  then consider n be Element of NAT such that
A3: (f1/*s).n=x by FUNCT_2:113;
  s.n in rng s by VALUED_0:28;
  then f1.(s.n) in dom f2 by A1,FUNCT_1:11;
  hence thesis by A1,A2,A3,FUNCT_2:108,XBOOLE_1:1;
end;
