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 Th45:
  for B being Element of Fin X holds FinUnion(B \/ {.i.}, f) =
  FinUnion(B,f) \/ f.i
proof
  let B be Element of Fin X;
A1: FinUnion A is associative by Th36;
  FinUnion A is idempotent & FinUnion A is commutative by Th34,Th35;
  hence FinUnion(B \/ {.i.}, f) = FinUnion A.(FinUnion(B,f), f.i) by A1,Th29
,Th38
    .= FinUnion(B,f) \/ f.i by Def4;
end;
