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 Th47:
  s = Class(EqCl ReductionRel H,p) & t = Class(EqCl ReductionRel H,q) implies
    s*t = Class(EqCl ReductionRel H,p^q)
proof
  set R = EqCl ReductionRel H;
  assume A1: s = Class(R,p) & t = Class(R,q);
  p in FreeAtoms(H)* & q in FreeAtoms(H)* by FINSEQ_1:def 11;
  then reconsider v=p, w=q as Element of FreeAtoms(H)*+^+<0> by MONOID_0:61;
  reconsider x=s, y=t as Element of Class R;
  thus s*t = Class(R,v*w) by A1, ALGSTR_4:def 4
    .= Class(R,p^q) by MONOID_0:62;
end;
