
theorem Th1:
  for f being Function, F being Function-yielding Function st f =
  union rng F holds dom f = union rng doms F
proof
  let f be Function;
  let F be Function-yielding Function;
  assume
A1: f = union rng F;
  thus dom f c= union rng doms F
  proof
    let x be object;
    assume x in dom f;
    then [x,f.x] in union rng F by A1,FUNCT_1:def 2;
    then consider g being set such that
A2: [x,f.x] in g and
A3: g in rng F by TARSKI:def 4;
    consider u being object such that
A4: u in dom F and
A5: g = F.u by A3,FUNCT_1:def 3;
    u in dom doms F by A4,A5,FUNCT_6:22;
    then
A6: (doms F).u in rng doms F by FUNCT_1:def 3;
    x in dom (F.u) by A2,A5,FUNCT_1:1;
    then x in (doms F).u by A4,FUNCT_6:22;
    hence thesis by A6,TARSKI:def 4;
  end;
  let x be object;
  assume x in union rng doms F;
  then consider A be set such that
A7: x in A and
A8: A in rng doms F by TARSKI:def 4;
  consider u being object such that
A9: u in dom doms F and
A10: A = (doms F).u by A8,FUNCT_1:def 3;
A11: u in dom F by A9,FUNCT_6:59;
  then
A12: F.u in rng F by FUNCT_1:def 3;
  consider g being Function such that
A13: g = F.u;
  A = dom (F.u) by A10,A11,FUNCT_6:22;
  then [x,g.x] in F.u by A7,A13,FUNCT_1:def 2;
  then [x,g.x] in f by A1,A12,TARSKI:def 4;
  hence thesis by FUNCT_1:1;
end;
