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 Th51:
  [* i, g *] * [* i, h *] = [* i, g*h *]
proof
  set R = EqCl ReductionRel(H);
  [i,g] in FreeAtoms(H) & [i,h] in FreeAtoms(H) by Th9;
  then reconsider s1 = <*[i,g]*>, s2 = <*[i,h]*> as FinSequence of FreeAtoms H
    by FINSEQ_1:74;
  s1^s2 in FreeAtoms(H)*;
  then A1: s1^s2 in the carrier of FreeAtoms(H)*+^+<0> by MONOID_0:61;
  A2: [<*[i,g],[i,h]*>, <*[i,g*h]*>] in ReductionRel(H) by Th27;
  ReductionRel(H) c= EqCl ReductionRel(H) by MSUALG_5:def 1;
  then [<*[i,g],[i,h]*>, <*[i,g*h]*>] in R by A2;
  then A3: [s1^s2, <*[i,g*h]*>] in R by FINSEQ_1:def 9;
  thus [* i,g *] * [* i,h *] = Class(R,s1^s2) by Th47
    .= [* i,g*h *] by A1, A3, EQREL_1:35;
end;
