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 Th75:
  a > 0 implies a #R (b+c) = a #R b * a #R c
proof
  consider sb being Rational_Sequence such that
A1: sb is convergent and
A2: b = lim sb and
  for n holds sb.n<=b by Th67;
  consider sc being Rational_Sequence such that
A3: sc is convergent and
A4: c = lim sc and
  for n holds sc.n<=c by Th67;
  assume
A5: a>0;
  then
A6: a #Q sb is convergent by A1,Th69;
A7: a #Q sc is convergent by A5,A3,Th69;
  reconsider s = sb + sc as Rational_Sequence;
A8: lim s = b + c by A1,A2,A3,A4,SEQ_2:6;
A9: now
    let n;
    thus a #Q s .n = a #Q (s.n) by Def5
      .= a #Q (sb.n + sc.n) by SEQ_1:7
      .= a #Q (sb.n) * a #Q (sc.n) by A5,Th53
      .= a #Q (sb.n) * (a #Q sc .n) by Def5
      .= (a #Q sb .n) * (a #Q sc .n) by Def5;
  end;
A10: s is convergent by A1,A3;
  then a #Q s is convergent by A5,Th69;
  hence a #R (b+c) = lim (a #Q s) by A5,A10,A8,Def6
    .= lim ((a #Q sb) (#) (a #Q sc)) by A9,SEQ_1:8
    .= (lim (a #Q sb)) * (lim (a #Q sc)) by A6,A7,SEQ_2:15
    .= a #R b * (lim (a #Q sc)) by A5,A1,A2,A6,Def6
    .= a #R b * a #R c by A5,A3,A4,A7,Def6;
end;
