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 Th90:
  a>0 & s1 is convergent & s2 is convergent & (for n holds s2.n =
  a #R (s1.n)) implies lim s2 = a #R (lim s1)
proof
  assume that
A1: a>0 and
A2: s1 is convergent and
A3: s2 is convergent and
A4: for n holds s2.n = a #R (s1.n);
  per cases;
  suppose
    a>=1;
    hence thesis by A2,A3,A4,Lm12;
  end;
  suppose
A5: a<1;
A6: now
      assume
A7:   lim s2 = 0;
      a #R ((lim s1) + 1) > 0 by A1,Th81;
      then consider n such that
A8:   for m st m>=n holds |.s2.m-0.|<a #R ((lim s1) +1) by A3,A7,SEQ_2:def 7;
      consider n1 being Nat such that
A9:   for m st m>=n1 holds |.s1.m-lim s1.|<1 by A2,SEQ_2:def 7;
      now
        let m such that
A10:    m>=n+n1;
        n+n1>=n1 by NAT_1:12;
        then m>=n1 by A10,XXREAL_0:2;
        then
A11:    |.s1.m-lim s1.|<1 by A9;
        s1.m-lim s1<=|.s1.m-lim s1.| by ABSVALUE:4;
        then s1.m-lim s1< 1 by A11,XXREAL_0:2;
        then
A12:    s1.m-lim s1 + lim s1 < lim s1 + 1 by XREAL_1:6;
        n+n1>=n by NAT_1:12;
        then m>=n by A10,XXREAL_0:2;
        then |.s2.m-0.|<a #R ((lim s1)+1) by A8;
        then
A13:    |.a #R (s1.m).|<a #R ((lim s1)+1) by A4;
        a #R (s1.m) > 0 by A1,Th81;
        then a #R (s1.m) < a #R ((lim s1)+1) by A13,ABSVALUE:def 1;
        hence contradiction by A1,A5,A12,Th84;
      end;
      hence contradiction;
    end;
A14: now
      let n;
      thus s2".n = (s2.n)" by VALUED_1:10
        .= (a #R (s1.n))" by A4
        .= 1/(a #R (s1.n))
        .= (1/a) #R (s1.n) by A1,Th79;
    end;
    a*(1/a)<=1*(1/a) by A1,A5,XREAL_1:64;
    then
A15: 1<=1/a by A1,XCMPLX_1:106;
A16: a #R (lim s1) <> 0 by A1,Th81;
    now
      let n;
      s2.n = a #R (s1.n) by A4;
      hence s2.n <> 0 by A1,Th81;
    end;
    then
A17: s2 is non-zero by SEQ_1:5;
    then
A18: lim s2" = (lim s2)" by A3,A6,SEQ_2:22;
    s2" is convergent by A3,A6,A17,SEQ_2:21;
    then (lim s2)" = (1/a) #R (lim s1) by A2,A15,A18,A14,Lm12
      .= 1/a #R (lim s1) by A1,Th79;
    then 1 = 1/a #R (lim s1) * lim s2 by A6,XCMPLX_0:def 7;
    then a #R (lim s1)= a #R (lim s1)*(1/a #R (lim s1) * lim s2)
      .= a #R (lim s1)*(1/a #R (lim s1)) * lim s2
      .= 1 * lim s2 by A16,XCMPLX_1:106;
    hence thesis;
  end;
end;
