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 Th70:
  ((nf s).len nf s)`1 <> ((nf t).1)`1 implies
    factorization(s * t) = factorization(s) ^ factorization(t)
proof
  reconsider f = uncurry injection H as ManySortedSet of FreeAtoms(H) by Lm5;
  assume ((nf s).len nf s)`1 <> ((nf t).1)`1;
  hence factorization(s * t) = f*((nf s) ^ nf t) by Th66
    .= factorization(s) ^ factorization(t) by FINSEQOP:9;
end;
