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
  a > 0 implies a #R b #R c = a #R (b * c)
proof
  consider s being Rational_Sequence such that
A1: s is convergent and
A2: c = lim s and
  for n holds s.n<=c by Th67;
A3: lim (b(#)s) = b*c by A1,A2,SEQ_2:8;
A4: b(#)s is convergent by A1;
  assume
A5: a>0;
  then
A6: (a #R b) #Q s is convergent by A1,Th69,Th81;
A7: now
    let n;
    thus (a #R b) #Q s .n = (a #R b) #Q (s.n) by Def5
      .= a #R (b*s.n) by A5,Lm11
      .= a #R ((b(#)s).n) by SEQ_1:9;
  end;
  a #R b > 0 by A5,Th81;
  then (a #R b) #R c = lim ((a #R b) #Q s) by A1,A2,A6,Def6;
  hence thesis by A5,A6,A4,A3,A7,Th90;
end;
