
theorem Th3:
  for x,y be Real st for c be Real st c > 0 & c < 1
  holds c*x >= y holds y <= 0
proof
  let x,y be Real;
  assume
A1: for c be Real st c > 0 & c < 1 holds c*x >= y;
  set ma = max(x,y);
  set mi = min(x,y);
  set c = mi/(2*ma);
  assume
A2: y > 0;
  then
A3: y*2 > y by XREAL_1:155;
  per cases;
  suppose
A4: x > 0;
    then
A5: mi > 0 & ma > 0 by A2,XXREAL_0:15,16;
    then mi/ma*2 > mi/ma by XREAL_1:155;
    then mi/ma > mi/ma/2 by XREAL_1:83;
    then
A6: mi/ma > mi/(ma*2) by XCMPLX_1:78;
    mi/ma <= 1 by A4,Th2;
    then c < 1 by A6,XXREAL_0:2;
    then
A7: c*x >= y by A1,A5;
    now
      per cases;
      suppose
        x >= y;
        then ma = x & mi = y by XXREAL_0:def 9,def 10;
        then c*x = y/2 by A4,XCMPLX_1:92;
        hence contradiction by A3,A7,XREAL_1:83;
      end;
      suppose
A8:     x < y;
        then ma = y & mi = x by XXREAL_0:def 9,def 10;
        then c*x < x/(2*y)*y by A4,A8,XREAL_1:98;
        then
A9:     c*x < x/2 by A2,XCMPLX_1:92;
A10:    y > y/2 by A3,XREAL_1:83;
        x/2 < y/2 by A8,XREAL_1:74;
        then c*x < y/2 by A9,XXREAL_0:2;
        hence contradiction by A7,A10,XXREAL_0:2;
      end;
    end;
    hence contradiction;
  end;
  suppose
    x <= 0;
    then 1/2*x <= 0;
    hence contradiction by A1,A2;
  end;
end;
