reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;

theorem Th69:
  for s being Rational_Sequence st s is convergent & a>0 holds a
  #Q s is convergent
proof
  let s be Rational_Sequence;
  assume that
A1: s is convergent and
A2: a>0;
  per cases;
  suppose
    a>=1;
    hence thesis by A1,Lm6;
  end;
  suppose
A3: a<1;
    then a/a<1/a by A2,XREAL_1:74;
    then 1<1/a by A2,XCMPLX_1:60;
    then
A4: (1/a) #Q s is convergent by A1,Lm6;
    s is bounded by A1;
    then consider d be Real such that
    0<d and
A5: for n holds |.s.n.|<d by SEQ_2:3;
    reconsider d as Real;
    consider m1 such that
A6: 2*d < m1 by SEQ_4:3;
    reconsider m1 as Rational;
A7: a #Q m1 > 0 by A2,Th52;
    now
      let c be Real;
      assume
A8:   c>0;
      then c* a #Q m1 > 0 by A7;
      then consider n such that
A9:   for m st n<=m holds |.((1/a) #Q s).m -((1/a) #Q s).n.|<c* a #Q
      m1 by A4,SEQ_4:41;
      take n;
      let m;
      assume m>=n;
      then
A10:  |.((1/a) #Q s).m -((1/a) #Q s).n.|<c* a #Q m1 by A9;
A11:  a #Q (s.m) <> 0 by A2,Th52;
A12:  a #Q (s.m+s.n) > 0 by A2,Th52;
      |.s.m.|<d by A5;
      then
A13:  |.s.m.|+|.s.n.|<d+d by A5,XREAL_1:8;
      |.s.m+s.n.|<=|.s.m.|+|.s.n.| by COMPLEX1:56;
      then |.s.m+s.n.|<d+d by A13,XXREAL_0:2;
      then |.s.m+s.n.|<m1 by A6,XXREAL_0:2;
      then
A14:  |.-(s.m+s.n).|<m1 by COMPLEX1:52;
      -(s.m+s.n)<=|.-(s.m+s.n).| by ABSVALUE:4;
      then -(s.m+s.n)<m1 by A14,XXREAL_0:2;
      then
A15:  m1-(-(s.m+s.n))>0 by XREAL_1:50;
A16:  a #Q (s.n) <> 0 by A2,Th52;
      |.((1/a) #Q s).m -((1/a) #Q s).n.| = |.(1/a) #Q (s.m) -((1/a) #Q
      s) . n.| by Def5
        .= |.(1/a) #Q (s.m) -(1/a) #Q (s.n).| by Def5
        .= |.1/a #Q (s.m) -(1/a) #Q (s.n).| by A2,Th57
        .= |.1/a #Q (s.m) -1/a #Q (s.n).| by A2,Th57
        .= |.(a #Q (s.m))" -1/a #Q (s.n).|
        .= |.(a #Q (s.m))" -(a #Q (s.n))".|
        .= |.a #Q (s.m) - a #Q (s.n).|/(|.a #Q (s.m).|*|.a #Q (s.n).|) by
A11,A16,SEQ_2:2
        .= |.a #Q (s.m) - a #Q (s.n).|/|.a #Q (s.m) * a #Q (s.n).| by
COMPLEX1:65
        .= |.a #Q (s.m) - a #Q (s.n).|/|.a #Q (s.m+s.n).| by A2,Th53
        .= |.a #Q (s.m) - a #Q (s.n).|/(a #Q (s.m+s.n)) by A12,ABSVALUE:def 1;
      then
A17:  (|.a #Q (s.m) - a #Q (s.n).|/a #Q (s.m+s.n))*a #Q (s.m+s.n) < c*a
      #Q m1 * a #Q (s.m+s.n) by A10,A12,XREAL_1:68;
      a #Q (s.m+s.n) <> 0 by A2,Th52;
      then
A18:  |.a #Q (s.m) - a #Q (s.n).|<c * a #Q m1 * a #Q (s.m+s.n) by A17,
XCMPLX_1:87;
      a #Q m1 * a #Q (s.m+s.n) = a #Q (m1+(s.m+s.n)) by A2,Th53;
      then c*(a #Q m1 * a #Q (s.m+s.n)) < 1*c by A2,A3,A8,A15,Th65,XREAL_1:68;
      then |.a #Q (s.m) - a #Q (s.n).|<c by A18,XXREAL_0:2;
      then |.(a #Q s).m - a #Q (s.n).| < c by Def5;
      hence |.(a #Q s).m - (a #Q s).n.| < c by Def5;
    end;
    hence thesis by SEQ_4:41;
  end;
end;
