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 Th49:
  [* i,1_(H.i) *] = 1_FreeProduct(H)
proof
  [i,1_(H.i)] in FreeAtoms(H) by Th9;
  then <*[i,1_(H.i)]*> is FinSequence of FreeAtoms(H) by FINSEQ_1:74;
  then <*[i,1_(H.i)]*> in FreeAtoms(H)* by FINSEQ_1:def 11;
  then A1: <*[i,1_(H.i)]*> in the carrier of FreeAtoms(H)*+^+<0>
    by MONOID_0:61;
  A2: [<*[i,1_(H.i)]*>,{}] in ReductionRel(H) by Th29;
  ReductionRel(H) c= EqCl ReductionRel(H) by MSUALG_5:def 1;
  then Class(EqCl ReductionRel H,{}) = [*i,1_(H.i)*] by A1, A2, EQREL_1:35;
  hence thesis by Th45;
end;
