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 Th59:
  s = Class(EqCl ReductionRel H,p) implies nf s = nf(p,ReductionRel H)
proof
  assume A1: s = Class(EqCl ReductionRel H,p);
  set q = nf(p,ReductionRel H);
  q is_a_normal_form_of p,ReductionRel H by REWRITE1:54;
  then A2: q is_a_normal_form_wrt ReductionRel H &
    ReductionRel H reduces p,q by REWRITE1:def 6;
  A3: p in FreeAtoms(H)* by FINSEQ_1:def 11;
  then p in field ReductionRel H by Th30;
  then q in field ReductionRel H by A2, REWRITE1:19;
  then A4: q in FreeAtoms(H)* by Th30;
  then A5: q in the carrier of FreeAtoms(H)*+^+<0> by MONOID_0:61;
  A6: p in the carrier of FreeAtoms(H)*+^+<0> by A3, MONOID_0:61;
  p,q are_convertible_wrt ReductionRel H by A2, REWRITE1:25;
  then [p,q] in EqCl ReductionRel H by A5, A6, MSUALG_6:41;
  then A7: q in s by A1, EQREL_1:18;
  q is FinSequence of FreeAtoms(H) by A4, FINSEQ_1:def 11;
  hence thesis by A2, A7, Def7;
end;
