reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;
reserve X,X1,X2 for Subset of A;
reserve Y for Subset of B;
reserve R,R1,R2 for Subset of [:A,B:];
reserve FR for Subset-Family of [:A,B:];

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