reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;
reserve X, Y for non empty set,
  F for (BinOp of Y),
  B for (Element of Fin X),
  f for Function of X,Y;
reserve A for set,
  x,y,z for Element of Fin A;
reserve X,Y for non empty set,
  A for set,
  f for (Function of X, Fin A),
  i,j,k for (Element of X);

theorem
  for B1,B2 being Element of Fin X holds FinUnion(B1 \/ B2, f) =
  FinUnion(B1,f) \/ FinUnion(B2,f)
proof
  let B1,B2 be Element of Fin X;
  thus FinUnion(B1 \/ B2, f) = union(f.:(B1 \/ B2)) by Th46
    .= union(f.:B1 \/ f .:B2) by RELAT_1:120
    .= union(f.:B1) \/ union(f.:B2) by ZFMISC_1:78
    .= FinUnion(B1,f) \/ union(f.:B2) by Th46
    .= FinUnion(B1,f) \/ FinUnion(B2,f) by Th46;
end;
