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

theorem Th2:
  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= dom f1 & rng s c= dom f1 \ X & rng(f1/*s) c= dom
  f2
proof
  let s be Real_Sequence,X be set such that
A1: rng s c=dom(f2*f1)\X;
  dom(f2*f1)\X c=dom(f2*f1) by XBOOLE_1:36;
  hence
A2: rng s c=dom(f2*f1) by A1;
A3: dom(f2*f1)c=dom f1 by RELAT_1:25;
  hence
A4: rng s c=dom f1 by A2;
  thus rng s c=dom f1\X
  proof
    let x be object;
    assume
A5: x in rng s;
    then not x in X by A1,XBOOLE_0:def 5;
    hence thesis by A4,A5,XBOOLE_0:def 5;
  end;
  let x be object;
  assume x in rng(f1/*s);
  then consider n be Element of NAT such that
A6: (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 A2,FUNCT_1:11;
  hence thesis by A2,A3,A6,FUNCT_2:108,XBOOLE_1:1;
end;
