reserve a,a1,b,b1,x,y for Real,
  F,G,H for FinSequence of REAL,
  i,j,k,n,m for Element of NAT,
  I for Subset of REAL,
  X for non empty set,
  x1,R,s for set;
reserve A for non empty closed_interval Subset of REAL;
reserve A, B for non empty closed_interval Subset of REAL;
reserve r for Real;
reserve D, D1, D2 for Division of A;
reserve f, g for Function of A,REAL;

theorem Th8:
  for f,g be PartFunc of X,REAL holds rng(f+g) c= rng f ++ rng g
proof
  let f,g be PartFunc of X,REAL;
    let y be object;
    assume y in rng(f+g);
    then consider x1 being object such that
A1: x1 in dom(f+g) and
A2: y=(f+g).x1 by FUNCT_1:def 3;
A3: dom(f+g)=dom f /\ dom g by VALUED_1:def 1;
    then x1 in dom f by A1,XBOOLE_0:def 4; then
A4: f.x1 in rng f by FUNCT_1:def 3;
    x1 in dom g by A1,A3,XBOOLE_0:def 4;
    then
A5: g.x1 in rng g by FUNCT_1:def 3;
    (f+g).x1=f.x1+g.x1 by A1,VALUED_1:def 1;
    hence thesis by A2,A4,A5,MEMBER_1:46;
end;
