reserve i,j,k,x for object;

theorem Th8:
  for A,B being functional set, A1 being Subset of A, B1 being
  Subset of B holds FuncComp(A1,B1) = FuncComp(A,B)|([:B1,A1:] qua set)
proof
  let A,B be functional set, A1 be Subset of A, B1 be Subset of B;
  set f = FuncComp(A,B)|([:B1,A1:] qua set);
A1: dom FuncComp(A,B) = [:B,A:] by PARTFUN1:def 2;
  then
A2: dom f = [:B1,A1:] by RELAT_1:62;
  then reconsider f as ManySortedFunction of [:B1,A1:] by PARTFUN1:def 2;
  f is compositional
  proof
    let i;
    assume
A3: i in dom f;
    then f.i = FuncComp(A,B).i by FUNCT_1:49;
    hence thesis by A1,A2,A3,Def9;
  end;
  hence thesis by Def10;
end;
