reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;
reserve F,G for Cardinal-Function;
reserve A,B for set;

theorem Th52:
  x in sproduct f implies x is PartFunc of dom f, union rng f
proof
  assume x in sproduct f;
  then consider g such that
A1: x = g and
A2: dom g c= dom f and
A3: for x being object st x in dom g holds g.x in f.x by Def9;
  rng g c= union rng f
  proof
    let y be object;
    assume y in rng g;
    then consider z being object such that
A4: z in dom g and
A5: y = g.z by FUNCT_1:def 3;
A6: g.z in f.z by A3,A4;
    f.z in rng f by A2,A4,FUNCT_1:def 3;
    hence thesis by A5,A6,TARSKI:def 4;
  end;
  hence thesis by A1,A2,RELSET_1:4;
end;
