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 Th43:
  for p1,p2,q1,q2 being FinSequence of FreeAtoms(H)
  st p1,q1 are_convertible_wrt ReductionRel(H) &
    p2,q2 are_convertible_wrt ReductionRel(H)
  holds p1^p2,q1^q2 are_convertible_wrt ReductionRel(H)
proof
  let p1,p2,q1,q2 be FinSequence of FreeAtoms(H);
  assume p1,q1 are_convertible_wrt ReductionRel(H) &
    p2,q2 are_convertible_wrt ReductionRel(H);
  then A1: p1,q1 are_convergent_wrt ReductionRel(H) &
    p2,q2 are_convergent_wrt ReductionRel(H) by REWRITE1:def 23;
  then consider c1 being object such that
    A2: ReductionRel(H) reduces p1,c1 & ReductionRel(H) reduces q1,c1
    by REWRITE1:def 7;
  p1 in FreeAtoms(H)* by FINSEQ_1:def 11;
  then p1 in field ReductionRel(H) by Th30;
  then c1 in field ReductionRel(H) by A2, REWRITE1:19;
  then c1 in FreeAtoms(H)* by Th30;
  then reconsider c1 as FinSequence of FreeAtoms(H) by FINSEQ_1:def 11;
  consider c2 being object such that
    A3: ReductionRel(H) reduces p2,c2 & ReductionRel(H) reduces q2,c2
    by A1, REWRITE1:def 7;
  p2 in FreeAtoms(H)* by FINSEQ_1:def 11;
  then p2 in field ReductionRel(H) by Th30;
  then c2 in field ReductionRel(H) by A3, REWRITE1:19;
  then c2 in FreeAtoms(H)* by Th30;
  then reconsider c2 as FinSequence of FreeAtoms(H) by FINSEQ_1:def 11;
  ReductionRel(H) reduces p1^p2,c1^c2 by A2, A3, Th41;
  then A4: p1^p2,c1^c2 are_convertible_wrt ReductionRel(H) by REWRITE1:25;
  ReductionRel(H) reduces q1^q2,c1^c2 by A2, A3, Th41;
  then c1^c2,q1^q2 are_convertible_wrt ReductionRel(H) by REWRITE1:25;
  hence thesis by A4, REWRITE1:30;
end;
