
theorem Th21:
  for X being non empty set, f,g being Function st rng g c= X
  holds (X-indexing f)*g = ((id X) +* f)*g
proof
  let X be non empty set, f,g be Function such that
A1: rng g c= X;
  rng g c= dom (X-indexing f) by A1,PARTFUN1:def 2;
  then
A2: dom ((X-indexing f)*g) = dom g by RELAT_1:27;
A3: now
    let x be object;
    assume
A4: x in dom g;
    then
A5: (((id X) +* f)*g).x = ((id X) +* f).(g.x) by FUNCT_1:13;
A6: g.x in rng g by A4,FUNCT_1:def 3;
    ((X-indexing f)*g).x = (X-indexing f).(g.x) by A4,FUNCT_1:13;
    hence ((X-indexing f)*g).x = (((id X) +* f)*g).x by A1,A5,A6,Th8;
  end;
  dom id X = X;
  then
A7: dom ((id X) +* f) = X \/ dom f by FUNCT_4:def 1;
  X c= X \/ dom f by XBOOLE_1:7;
  then dom (((id X) +* f)*g) = dom g by A1,A7,RELAT_1:27,XBOOLE_1:1;
  hence thesis by A2,A3;
end;
