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 Th27:
  for f being Function-yielding Function st x in dom <:f:>
  for g st g in rng f holds x in dom g
proof let f be Function-yielding Function;
  assume
A1: x in dom <:f:>;
  let g;
  assume g in rng f;
  then consider y being object such that
A2: y in dom f & g = f.y by FUNCT_1:def 3;
  y in dom doms f & (doms f).y = dom g by A2,Th18;
  then dom g in rng doms f by FUNCT_1:def 3;
  then
A3: meet rng doms f c= dom g by SETFAM_1:3;
  meet doms f = dom <:f:> by Th25;
  hence thesis by A1,A3;
end;
