reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th31:
  for x,y being Real for r being positive Real ex
  w,v being Rational st |[w,v]| in Ball(|[x,y]|,r) & |[w,v]| <> |[x,y]|
proof
  let x,y be Real;
  let r be positive Real;
  x < x+r/2 by XREAL_1:39;
  then consider w being Rational such that
A1: x < w and
A2: w < x+r/2 by RAT_1:7;
A3: w-x > 0 by A1,XREAL_1:50;
  y < y+r/2 by XREAL_1:39;
  then consider v being Rational such that
A4: y < v and
A5: v < y+r/2 by RAT_1:7;
A6: v-y > 0 by A4,XREAL_1:50;
  |[w,v]|-|[x,v]| = |[w-x,v-v]| by EUCLID:62;
  then |.|[w,v]|-|[x,v]|.| = |.w-x.| by TOPREAL6:23;
  then |.|[w,v]|-|[x,v]|.| = w-x by A3,ABSVALUE:def 1;
  then
A7: |.|[w,v]|-|[x,v]|.| < x+r/2-x by A2,XREAL_1:9;
  take w,v;
A8: |[x,v]|-|[x,y]| = |[x-x,v-y]| by EUCLID:62;
A9: |.|[w,v]|-|[x,y]|.| <= |.|[w,v]|-|[x,v]|.|+|.|[x,v]|-|[x,y]|.| by
TOPRNS_1:34;
  |.|[x,v]|-|[x,y]|.| = |.v-y.| by A8,TOPREAL6:23;
  then |.|[x,v]|-|[x,y]|.| = v-y by A6,ABSVALUE:def 1;
  then |.|[x,v]|-|[x,y]|.| <= y+r/2-y by A5,XREAL_1:9;
  then |.|[w,v]|-|[x,v]|.|+|.|[x,v]|-|[x,y]|.| < x+r/2-x+(y+r/2-y) by A7,
XREAL_1:8;
  then |.|[w,v]|-|[x,y]|.| < r by A9,XXREAL_0:2;
  hence |[w,v]| in Ball(|[x,y]|,r) by TOPREAL9:7;
  thus thesis by A4,SPPOL_2:1;
end;
