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