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 Th78:
  a > 0 & b > 0 implies (a * b) #R c = a #R c * b #R c
proof
  assume that
A1: a>0 and
A2: b>0;
A3: a*b > 0 by A1,A2;
  consider s1 being Rational_Sequence such that
A4: s1 is convergent and
A5: c = lim s1 and
  for n holds s1.n>=c by Th68;
A6: a #Q s1 is convergent by A1,A4,Th69;
A7: b #Q s1 is convergent by A2,A4,Th69;
  now
    let n;
    thus (a*b) #Q s1 .n = (a*b) #Q (s1.n) by Def5
      .= a #Q (s1.n) * b #Q (s1.n) by A1,A2,Th56
      .= (a #Q s1 .n) * b #Q (s1.n) by Def5
      .= (a #Q s1 .n) * (b #Q s1 .n) by Def5;
  end;
  then
A8: (a*b) #Q s1 = a #Q s1 (#) b #Q s1 by SEQ_1:8;
  then (a*b) #Q s1 is convergent by A6,A7;
  hence (a*b) #R c = lim ((a *b) #Q s1) by A3,A4,A5,Def6
    .= lim (a #Q s1) * lim (b #Q s1) by A6,A7,A8,SEQ_2:15
    .= a #R c * lim (b #Q s1) by A1,A4,A5,A6,Def6
    .= a #R c * b #R c by A2,A4,A5,A7,Def6;
end;
