
theorem Th3:
  for F being set, A being FinSequence of bool F, J being set holds
  union (A, J) c= F
proof
  let F be set, A be FinSequence of bool F, J be set;
  let x be object;
  assume x in union (A, J);
  then consider j being set such that
  j in J and
A1: j in dom A and
A2: x in A.j by Def1;
  rng A c= bool F & A.j in rng A by A1,FINSEQ_1:def 4,FUNCT_1:3;
  hence thesis by A2;
end;
