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;

theorem Th32:
  for p,q being FinSequence of FreeAtoms(G), g,h being Element of G.i
  st g*h = 1_(G.i) holds ReductionRel(G) reduces p^<*[i,g],[i,h]*>^q, p^q
proof
  let p,q be FinSequence of FreeAtoms(G), g,h be Element of G.i;
  assume A1: g*h = 1_(G.i);
  [p^<*[i,g],[i,h]*>^q, p^<*[i,g*h]*>^q] in ReductionRel(G) by Th26;
  then A2: ReductionRel(G) reduces p^<*[i,g],[i,h]*>^q, p^<*[i,g*h]*>^q
    by REWRITE1:15;
  [p^<*[i,g*h]*>^q, p^q] in ReductionRel(G) by A1, Th28;
  then ReductionRel(G) reduces p^<*[i,g*h]*>^q, p^q by REWRITE1:15;
  hence thesis by A2, REWRITE1:16;
end;
