reserve a,b,r,g for Real;

theorem Th2:
  for a,b,g1,M be Real st a < b & 0 < g1 & 0 < M holds ex r
  st a < r & r < b & (r - a)*M < g1
proof
  let a,b,g1,M be Real such that
A1: a < b and
A2: 0 < g1 & 0 < M;
  (-b) < (-a) by A1,XREAL_1:24;
  then consider r1 be Real such that
A3: (-b) < r1 & r1 < (-a) and
A4: ((-a) - r1)*M < g1 by A2,Th1;
  reconsider r= -r1 as Real;
  take r;
  -(-b) > -r1 & -r1 > -(-a) by A3,XREAL_1:24;
  hence thesis by A4;
end;
