reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem Th51:
  for f,g being Function holds rng <:f,g:> c= [:rng f,rng g:]
proof
  let f,g be Function;
  let q be object;
  assume q in rng <:f,g:>;
  then consider x being object such that
A1: x in dom <:f,g:> and
A2: q = <:f,g:>.x by FUNCT_1:def 3;
A3: x in dom f /\ dom g by A1,Def7;
  then x in dom f by XBOOLE_0:def 4;
  then
A4: f.x in rng f by FUNCT_1:def 3;
  x in dom g by A3,XBOOLE_0:def 4;
  then
A5: g.x in rng g by FUNCT_1:def 3;
  q = [f.x,g.x] by A1,A2,Def7;
  hence thesis by A4,A5,ZFMISC_1:87;
end;
