reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem
  y>x & x>=0 & m>=0 implies x/y <= (x+m)/(y+m)
proof
  assume that
A1: y>x and
A2: x>=0 and
A3: m>=0;
  y-x>0 by A1,XREAL_1:50;
  then m*(y-x)>=0 by A3;
  then (y*(x+m)-x*(m+y))/(y*(y+m))>=0 by A1,A2,A3;
  then (x+m)/(y+m)-x/y>=0 by A1,A2,A3,XCMPLX_1:130;
  then (x+m)/(y+m)-x/y+x/y>=0+x/y by XREAL_1:6;
  hence thesis;
end;
