theorem Th44:
  (repeat(Relax(n)*findmin(n))).i.f, (repeat(Relax(n)*findmin(n)))
  .(i+1).f equal_at 3*n+1,n*n+3*n
proof
  set R=Relax(n), M=findmin(n), ff=(repeat (R*M)).i.f;
  set Fi1=(repeat (R*M)).(i+1).f;
A1: now
    let k;
    assume that
A2: k in dom ff and
A3: 3*n+1 <= k and
A4: k <= n*n+3*n;
A5: k > 3*n by A3,NAT_1:13;
A6: k in dom (M.ff) by A2,Th33;
A7: k < n*n+3*n+1 by A4,NAT_1:13;
    3*n >= n by Lm6;
    then
A8: k > n by A5,XXREAL_0:2;
    thus Fi1.k=(R.(M.ff)).k by Th22
      .=(M.ff).k by A5,A6,Th36
      .=ff.k by A7,A8,Th31;
  end;
  dom (Fi1) = dom ff by Th37;
  hence thesis by A1;
end;
