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 Th45:
  for I being set, H being Group-like associative multMagma-Family of I
  holds 1_FreeProduct(H) = Class(EqCl ReductionRel H,{})
proof
  let I be set, H be Group-like associative multMagma-Family of I;
  thus 1_FreeProduct(H)
     = Class(EqCl ReductionRel H,1_(FreeAtoms(H)*+^+<0>)) by ALGSTR_4:49
    .= Class(EqCl ReductionRel H,{}) by MONOID_0:61;
end;
