
theorem Th1:
  for x,y be Real st 0<=x & x<y ex s0 be Real st 0<s0
  & x < y/(1+s0) & y/(1+s0) < y
proof
  let x,y be Real;
  assume that
A1: 0<=x and
A2: x<y;
  ex s0 be Real st 0<s0 & x < y/(1+s0) & y/(1+s0) < y
  proof
    consider s be Real such that
A3: x < s and
A4: s < y by A2,XREAL_1:5;
    now
      per cases by A1;
      case
A5:     0=x;
        set s0=s;
        y < y*(1+s0) by A1,A3,A4,XREAL_1:29,155;
        hence 0<s0 & x < y/(1+s0) & y/(1+s0) < y by A3,A4,A5,XREAL_1:83,139;
      end;
      case
A6:     0<x;
        set s0=s /x - 1;
A7:     (1+s0)"=1/(1+s0) by XCMPLX_1:215;
        then
A8:     s*(1+s0)"= (s*1)/(1+s0) by XCMPLX_1:74;
        (1+s0)*x=(x/x)*s by XCMPLX_1:75;
        then (1+s0)*x=1*s by A6,XCMPLX_1:60;
        then
A9:     (1+s0)"*(1+s0)*x=s*(1+s0)";
        0 < 1+s0 by A3,A6,XREAL_1:187;
        then
A10:    (1+s0)"*(1+s0)=1 by A7,XCMPLX_1:106;
A11:    1< 1+s0 by A3,A6,XREAL_1:187;
        then y < y*(1+s0) by A1,A2,XREAL_1:155;
        hence 0<s0 & x < y/(1+s0) & y/(1+s0) < y by A4,A11,A9,A10,A8,XREAL_1:50
,74,83;
      end;
    end;
    hence thesis;
  end;
  hence thesis;
end;
