reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem
  for x being object, f being Function-yielding Function st
   g in rng f &
   for g st g in rng f holds x in dom g
 holds x in dom <:f :>
proof let x be object, f be Function-yielding Function;
  assume that
A1: g in rng f and
A2: for g st g in rng f holds x in dom g;
  ex y being object st y in dom f & g = f.y by A1,FUNCT_1:def 3;
  then
A3: doms f <> {} by Th18;
A4: now
    let y be object;
    assume y in dom doms f;
    then
A5: y in dom f by Def1;
    reconsider g = f.y as Function;
A6: y in dom f by A5;
    then g in rng f by FUNCT_1:def 3;
    then x in dom g by A2;
    hence x in (doms f).y by A6,Th18;
  end;
  dom <:f:> = meet doms f by Th25;
  hence thesis by A3,A4,Th21;
end;
