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

theorem Th78:
  for x being Real, y being non negative Real for
r,a being positive Real st a <= 1 holds +(x,y,r)"[.0,a.[ = Ball(|[x,y
  ]|,r*a) /\ y>=0-plane
proof
  let x be Real;
  let y be non negative Real;
  let r,a be positive Real;
  set f = +(x,y,r);
  assume
A1: a <= 1;
  then r*a <= r*1 by XREAL_1:64;
  then
A2: Ball(|[x,y]|,r*a) c= Ball(|[x,y]|,r) by JORDAN:18;
  thus +(x,y,r)"[.0,a.[ c= Ball(|[x,y]|,r*a) /\ y>=0-plane
  proof
    let u be object;
    assume
A3: u in f"[.0,a.[;
    then reconsider p = u as Point of Niemytzki-plane;
    p in y>=0-plane by Lm1;
    then reconsider q = p as Element of TOP-REAL 2;
    f.p in [.0,a.[ by A3,FUNCT_2:38;
    then
A4: f.p < a by XXREAL_1:3;
A5: p = |[q`1,q`2]| by EUCLID:53;
    then
A6: q`2 >= 0 by Lm1,Th18;
    then p in Ball(|[x,y]|,r) by A4,A1,A5,Def6;
    then f.p = |.|[x,y]|-q.|/r by A5,A6,Def6;
    then
A7: |.|[x,y]|-q.| < r*a by A4,XREAL_1:77;
    |.|[x,y]|-q.| = |.q-|[x,y]|.| by TOPRNS_1:27;
    then p in Ball(|[x,y]|,r*a) by A7,TOPREAL9:7;
    hence thesis by Lm1,XBOOLE_0:def 4;
  end;
  let u be object;
  assume
A8: u in Ball(|[x,y]|,r*a) /\ y>=0-plane;
  then reconsider p = u as Point of Niemytzki-plane by Lm1,XBOOLE_0:def 4;
  reconsider q = p as Element of TOP-REAL 2 by A8;
A9: u in Ball(|[x,y]|,r*a) by A8,XBOOLE_0:def 4;
  then
A10: |.q-|[x,y]|.| < r*a by TOPREAL9:7;
A11: |.|[x,y]|-q.| = |.q-|[x,y]|.| by TOPRNS_1:27;
A12: p = |[q`1,q`2]| by EUCLID:53;
  u in y>=0-plane by A8,XBOOLE_0:def 4;
  then q`2 >= 0 by A12,Th18;
  then
A13: f.p = |.|[x,y]|-q.|/r by A2,A9,A12,Def6;
  then r*(f.p) = |.|[x,y]|-q.| by XCMPLX_1:87;
  then f.p < a by A10,A11,XREAL_1:64;
  then f.p in [.0,a.[ by A13,XXREAL_1:3;
  hence thesis by FUNCT_2:38;
end;
