
theorem
  for f being non-empty Function, X being set, i being object
  st i in dom f & product(f +* (i,X)) c= product f holds X c= f.i
proof
  let f be non-empty Function, X be set, i be object;
  assume A1: i in dom f & product(f +* (i,X)) c= product f;
    let x be object;
    assume A2: x in X;
    set g = the Element of product f;
    a3: g +* (i,x) in product(f +* (i,X)) by A1, A2, Th37;
    i in dom g by A1, CARD_3:9;
    then (g +* (i,x)).i = x by FUNCT_7:31;
    hence thesis by A1, a3, CARD_3:9;
end;
