reserve r,r1,r2, s,x for Real,
  i for Integer;

theorem Th31:
  0 <= r & r < 3/2*PI implies sin r > -1
proof
  assume that
A1: 0 <= r and
A2: r < 3/2*PI and
A3: sin r <= -1;
A4: sin r >= -1 by Th3;
  r <= 2*PI by A2,Lm9,XXREAL_0:2;
  hence thesis by A1,A2,A3,A4,Th27,XXREAL_0:1;
end;
