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
  product f <> {} implies
  (g in sproduct f iff ex h st h in product f & g c= h)
proof
  assume
A1: product f <> {};
  thus g in sproduct f implies ex h st h in product f & g c= h
  proof
    assume
A2: g in sproduct f;
    set k = the Element of product f;
    reconsider k as Function;
    take k +* g;
    thus k +* g in product f by A1,A2,Th53;
    thus thesis by FUNCT_4:25;
  end;
  given h such that
A3: h in product f and
A4: g c= h;
A5: dom h = dom f by A3,Th9;
A6: dom g c= dom h by A4,RELAT_1:11;
  now
    let x be object;
    assume
A7: x in dom g;
    then g.x = h.x by A4,GRFUNC_1:2;
    hence g.x in f.x by A3,A5,A6,A7,Th9;
  end;
  hence thesis by A5,A6,Def9;
end;
