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
  [* i, g *]" = [* i, g" *]
proof
  [* i,g *] * [* i,g" *] = [* i,g*g" *] by Th51
    .= [* i,1_(H.i) *] by GROUP_1:def 5
    .= 1_FreeProduct(H) by Th49;
  hence thesis by GROUP_1:12;
end;
