reserve a,b,r,g for Real;

theorem Th1:
  for a,b,g1,M be Real st a < b & 0 < g1 & 0 < M holds ex r
  st a < r & r < b & (b - r)*M < g1
proof
  let a,b,g1,M be Real such that
A1: a < b and
A2: 0 < g1 and
A3: 0 < M;
  set r3 = max(a,b - g1/M);
  b - g1/M < b by A2,A3,XREAL_1:44,139;
  then r3 < b by A1,XXREAL_0:16;
  then consider q be Real such that
A4: r3 < q and
A5: q < b by XREAL_1:5;
  reconsider r = q as Real;
  take r;
  b - g1/M <= r3 by XXREAL_0:25;
  then b - g1/M < r by A4,XXREAL_0:2;
  then b - g1/M - (r - g1/M) < r - (r - g1/M) by XREAL_1:14;
  then (b - r)*1 < g1/M;
  then a <= r3 & (b - r)*M < g1/1 by A3,XREAL_1:111,XXREAL_0:25;
  hence thesis by A4,A5,XXREAL_0:2;
end;
