reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;

theorem
  for A being non empty set, B being set, X being Subset of A,
  R being Relation of A,B holds
  {R.:x where x is Element of A: x in X} is Subset-Family of B
proof
  let A be non empty set, B be set, X be Subset of A, R be Relation of A,B;
  deffunc F(Element of A) = R.:$1;
  defpred P[set] means $1 in X;
  set Y = { F(x) where x is Element of A: P[x]};
  thus Y is Subset-Family of B from DOMAIN_1:sch 8;
end;
