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 Th64:
 for x being set holds
  x c= h & h in sproduct f implies x in sproduct f
proof let x be set;
  assume that
A1: x c= h and
A2: h in sproduct f;
  reconsider g = x as Function by A1;
A3: dom g c= dom h by A1,GRFUNC_1:2;
  dom h c= dom f by A2,Th49;
  then
A4: dom g c= dom f by A3;
  now
    let x be object;
    assume
A5: x in dom g;
    then h.x in f.x by A2,A3,Th49;
    hence g.x in f.x by A1,A5,GRFUNC_1:2;
  end;
  hence thesis by A4,Def9;
end;
