reserve a,b,c,d for positive Real,
  m,u,w,x,y,z for Real,
  n,k for Nat,
  s,s1 for Real_Sequence;

theorem Th4:
  a<b implies a/b<sqrt(a/b)
proof
  b^2>0;
  then b|^2>0 by NEWTON:81;
  then b|^2*b>0;
  then
A1: b|^(2+1)>0 by NEWTON:6;
  assume a<b;
  then a-b<0 by XREAL_1:49;
  then (a*b)*(a-b)<0;
  then a^2*b-a*b^2+(a*b^2)<0+(a*b^2) by XREAL_1:8;
  then (a^2*b)/(b|^(2+1))<(a*b^2)/(b|^(2+1)) by A1,XREAL_1:74;
  then (a^2*b)/(b|^2*b)<(a*b^2)/(b|^(2+1)) by NEWTON:6;
  then (a^2*b)/(b|^2*b)<(a*b^2)/(b|^2*b) by NEWTON:6;
  then (a^2*b)/(b^2*b)<(a*b^2)/(b|^2*b) by NEWTON:81;
  then (a^2*b)/(b^2*b)<(a*b^2)/(b^2*b) by NEWTON:81;
  then a^2/b^2<(a*b^2)/(b*b^2) by XCMPLX_1:91;
  then a^2/b^2<a/b by XCMPLX_1:91;
  then (a/b)^2<a/b by XCMPLX_1:76;
  then sqrt((a/b)^2)<sqrt(a/b) by SQUARE_1:27;
  hence thesis by SQUARE_1:22;
end;
