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 Th70:
  for s1,s2 being Rational_Sequence, a st s1 is convergent & s2 is
  convergent & lim s1 = lim s2 & a>0 holds a #Q s1 is convergent & a #Q s2 is
  convergent & lim a #Q s1 = lim a #Q s2
proof
  let s1,s2 be Rational_Sequence;
  let a;
  assume that
A1: s1 is convergent and
A2: s2 is convergent and
A3: lim s1 = lim s2 and
A4: a>0;
  thus
A5: a #Q s1 is convergent by A1,A4,Th69;
  s2 is bounded by A2;
  then consider e be Real such that
  0<e and
A6: for n holds |.s2.n.|<e by SEQ_2:3;
  s1 is bounded by A1;
  then consider d be Real such that
  0<d and
A7: for n holds |.s1.n.|<d by SEQ_2:3;
  consider m1 such that
A8: d+e < m1 by SEQ_4:3;
  thus
A9: a #Q s2 is convergent by A2,A4,Th69;
  reconsider m1 as Rational;
A10: a #Q m1 > 0 by A4,Th52;
  per cases;
  suppose
    a>=1;
    hence thesis by A1,A2,A3,Lm7;
  end;
  suppose
A11: a<1;
    then a/a<1/a by A4,XREAL_1:74;
    then 1<1/a by A4,XCMPLX_1:60;
    then
A12: lim (1/a) #Q s1 = lim (1/a) #Q s2 by A1,A2,A3,Lm7;
A13: (1/a) #Q s2 is convergent by A2,A4,Th69;
A14: (1/a) #Q s1 is convergent by A1,A4,Th69;
    then
A15: (1/a) #Q s1 - (1/a) #Q s2 is convergent by A13;
A16: lim ((1/a) #Q s1 - (1/a) #Q s2) = lim (1/a) #Q s1 - lim (1/a) #Q s2
    by A14,A13,SEQ_2:12
      .= 0 by A12;
A17: now
      let c be Real;
      assume
A18:  c>0;
      then c* a #Q m1 > 0 by A10;
      then consider n such that
A19:  for m st n<=m holds |.((1/a) #Q s1 - (1/a) #Q s2).m-0.|<c*a #Q
      m1 by A15,A16,SEQ_2:def 7;
      take n;
      let m;
      assume m>=n;
      then |.((1/a) #Q s1 - (1/a) #Q s2).m-0.|<c*a #Q m1 by A19;
      then
A20:  |.((1/a) #Q s1).m - ((1/a) #Q s2).m.|<c*a #Q m1 by RFUNCT_2:1;
A21:  a #Q (s1.m) <> 0 by A4,Th52;
      |.s1.m.|<d by A7;
      then
A22:  |.s1.m.|+|.s2.m.|<d+e by A6,XREAL_1:8;
      |.s1.m+s2.m.|<=|.s1.m.|+|.s2.m.| by COMPLEX1:56;
      then |.s1.m+s2.m.|<d+e by A22,XXREAL_0:2;
      then |.s1.m+s2.m.|<m1 by A8,XXREAL_0:2;
      then
A23:  |.-(s1.m+s2.m).|<m1 by COMPLEX1:52;
      -(s1.m+s2.m)<=|.-(s1.m+s2.m).| by ABSVALUE:4;
      then -(s1.m+s2.m)<m1 by A23,XXREAL_0:2;
      then
A24:  m1-(-(s1.m+s2.m))>0 by XREAL_1:50;
A25:  a #Q (s2.m) <> 0 by A4,Th52;
A26:  a #Q (s1.m+s2.m) > 0 by A4,Th52;
      |.((1/a) #Q s1).m -((1/a) #Q s2).m.| = |.(1/a) #Q (s1.m) -((1/a)
      #Q s2).m.| by Def5
        .= |.(1/a) #Q (s1.m) -(1/a) #Q (s2.m).| by Def5
        .= |.1/a #Q (s1.m) -(1/a) #Q (s2.m).| by A4,Th57
        .= |.1/a #Q (s1.m) -1/a #Q (s2.m).| by A4,Th57
        .= |.(a #Q (s1.m))" -1/a #Q (s2.m).|
        .= |.(a #Q (s1.m))" -(a #Q (s2.m))".|
        .= |.a #Q (s1.m) - a #Q (s2.m).|/(|.a #Q (s1.m).|*|.a #Q (s2.m).|
      ) by A21,A25,SEQ_2:2
        .= |.a #Q (s1.m) - a #Q (s2.m).|/(|.a #Q (s1.m) * a #Q (s2.m).|)
      by COMPLEX1:65
        .= |.a #Q (s1.m) - a #Q (s2.m).|/|.a #Q (s1.m+s2.m).| by A4,Th53
        .= |.a #Q (s1.m) - a #Q (s2.m).|/(a #Q (s1.m+s2.m)) by A26,
ABSVALUE:def 1;
      then
A27:  (|.a #Q (s1.m) - a #Q (s2.m).|/a #Q (s1.m+s2.m))*a #Q (s1.m+s2.m)
      < c*a #Q m1 * a #Q (s1.m+s2.m) by A20,A26,XREAL_1:68;
      a #Q (s1.m+s2.m) <> 0 by A4,Th52;
      then
A28:  |.a #Q (s1.m) - a #Q (s2.m).|<c * a #Q m1 * a #Q (s1.m+s2.m) by A27,
XCMPLX_1:87;
      a #Q m1 * a #Q (s1.m+s2.m) = a #Q (m1+(s1.m+s2.m)) by A4,Th53;
      then c*(a #Q m1 * a #Q (s1.m+s2.m)) < 1*c by A4,A11,A18,A24,Th65,
XREAL_1:68;
      then |.a #Q (s1.m) - a #Q (s2.m).|<c by A28,XXREAL_0:2;
      then |.(a #Q s1).m - a #Q (s2.m).| < c by Def5;
      then |.(a #Q s1).m - (a #Q s2).m.| < c by Def5;
      hence |.((a #Q s1) - (a #Q s2)).m - 0.| < c by RFUNCT_2:1;
    end;
    then ((a #Q s1) - (a #Q s2)) is convergent by SEQ_2:def 6;
    then lim((a #Q s1) - (a #Q s2)) = 0 by A17,SEQ_2:def 7;
    then 0 = lim (a #Q s1)-lim (a #Q s2) by A5,A9,SEQ_2:12;
    hence thesis;
  end;
end;
