reserve x1, x2, x3, x4, x5, x6, x7 for set;

theorem Th21:
  for a being Rational, b being irrational Real st
  a <> 0 holds a * b is irrational
proof
  let a be Rational, b be irrational Real;
  assume
A1: a <> 0;
  assume a * b is rational;
  then consider m, n being Integer such that
  n > 0 and
A2: a * b = m / n by RAT_1:2;
  consider m1, n1 being Integer such that
  n1 > 0 and
A3: a = m1 / n1 by RAT_1:2;
  a * b / a = (m * n1) / (n * m1) by A2,A3,XCMPLX_1:84;
  hence thesis by A1,XCMPLX_1:89;
end;
