theorem Th53:
  for f,g being ManySortedSet of I holds dom commute <* <: f,g :> *> = I
proof
  let f,g be ManySortedSet of I;
  A1: dom <: f,g :> = I by Lm3;
  thus dom commute <* <: f,g :> *>
     = proj2 dom uncurry <* <: f,g :> *> by FUNCT_5:23
    .= proj2 dom uncurry { [1,<: f,g :>] } by FINSEQ_1:def 5
    .= proj2 dom uncurry (1 .--> <: f,g :>) by FUNCT_4:82
    .= proj2 dom uncurry ({1} --> <: f,g :>) by FUNCOP_1:def 9
    .= rng [:{1},dom <: f,g :>:] by FUNCT_6:8
    .= I by A1, RELAT_1:160;
end;
