
theorem Th24:
  for f being Function, g being rng-retract of f for x being set
  st x in rng f holds g.x in dom f & f.(g.x) = x
proof
  let f be Function, g be rng-retract of f, x be set such that
A1: x in rng f;
A2: rng g c= dom f by Th23;
A3: dom g = rng f by Def2;
  then g.x in rng g by A1,FUNCT_1:def 3;
  hence g.x in dom f by A2;
  thus f.(g.x) = (f*g).x by A1,A3,FUNCT_1:13
    .= (id rng f).x by Def2
    .= x by A1,FUNCT_1:18;
end;
