
theorem Th54:
  for I being non empty set
  for J being TopStruct-yielding non-Empty ManySortedSet of I
  for f being one-to-one I-valued Function
  for i being Element of I st i in rng f
  holds product_basis_selector(J,f).i = proj(J,i).:(f".i)
proof
  let I be non empty set;
  let J be TopStruct-yielding non-Empty ManySortedSet of I;
  let f be one-to-one I-valued Function;
  let i be Element of I;
  assume A1: i in rng f;
  then A2: i in dom(f") by FUNCT_1:33;
  thus product_basis_selector(J,f).i
     = ((ProjMap J).:.:(I-indexing(f"))).i by A1, FUNCT_1:49
    .= ((ProjMap J).i).:((I-indexing(f")).i) by PBOOLE:def 20
    .= proj(J,i).:((I-indexing(f")).i) by Th53
    .= proj(J,i).:(f".i) by A2, ALGSPEC1:9;
end;
