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 Th67:
  for f,g being Function holds rng [:f,g:] = [:rng f,rng g:]
proof
  let f,g be Function;
  for q being object holds q in rng [:f,g:] iff q in [:rng f,rng g:]
  proof let q be object;
A1: dom [:f,g:]=[:dom f,dom g:] by Def8;
    thus q in rng [:f,g:] implies q in [:rng f,rng g:]
    proof
      assume q in rng [:f,g:];
      then consider p being object such that
A2:   p in dom [:f,g:] and
A3:   q = [:f,g:].p by FUNCT_1:def 3;
      p in [:dom f, dom g:] by A2,Def8;
      then consider x,y being object such that
A4:   x in dom f & y in dom g and
A5:   p = [x,y] by ZFMISC_1:def 2;
A6:   f.x in rng f & g.y in rng g by A4,FUNCT_1:def 3;
      q = [:f,g:].(x,y) by A3,A5
        .= [f.x,g.y] by A4,Def8;
      hence thesis by A6,ZFMISC_1:87;
    end;
    assume q in [:rng f,rng g:];
    then consider y1,y2 being object such that
A7: y1 in rng f and
A8: y2 in rng g and
A9: q = [y1,y2] by ZFMISC_1:def 2;
    consider x2 being object such that
A10: x2 in dom g and
A11: y2 = g.x2 by A8,FUNCT_1:def 3;
    consider x1 being object such that
A12: x1 in dom f and
A13: y1 = f.x1 by A7,FUNCT_1:def 3;
A14: [x1,x2] in [:dom f,dom g:] by A12,A10,ZFMISC_1:87;
    [:f,g:].(x1,x2)=q by A9,A12,A13,A10,A11,Def8;
    hence thesis by A14,A1,FUNCT_1:def 3;
  end;
  hence thesis by TARSKI:2;
end;
