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

theorem Th30:
  0 <= r & r < PI/2 implies sin r < 1
proof
  assume that
A1: 0 <= r and
A2: r < PI/2 and
A3: sin r >= 1;
A4: sin r <= 1 by Th4;
  1/2*PI <= 2*PI by XREAL_1:64;
  then r < 2*PI by A2,XXREAL_0:2;
  hence thesis by A1,A2,A3,A4,Th28,XXREAL_0:1;
end;
