reserve x,y,z for object, X for set, I for non empty set, i,j for Element of I,
    M0 for multMagma-yielding Function,
    M for non empty multMagma-yielding Function,
    M1, M2, M3 for non empty multMagma,
    G for Group-like multMagma-Family of I,
    H for Group-like associative multMagma-Family of I;
reserve p, q for FinSequence of FreeAtoms(H), g,h for Element of H.i,
  k for Nat;
reserve s,t for Element of FreeProduct(H);

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;
