let V be RealUnitarySpace; :: thesis: for v1, v2 being Point of V
for r1, r2 being Real ex u being Point of V ex r being Real st (Ball v1,r1) \/ (Ball v2,r2) c= Ball u,r
let v1, v2 be Point of V; :: thesis: for r1, r2 being Real ex u being Point of V ex r being Real st (Ball v1,r1) \/ (Ball v2,r2) c= Ball u,r
let r1, r2 be Real; :: thesis: ex u being Point of V ex r being Real st (Ball v1,r1) \/ (Ball v2,r2) c= Ball u,r
reconsider u = v1 as Point of V ;
reconsider r = ((abs r1) + (abs r2)) + (dist v1,v2) as Real ;
take u ; :: thesis: ex r being Real st (Ball v1,r1) \/ (Ball v2,r2) c= Ball u,r
take r ; :: thesis: (Ball v1,r1) \/ (Ball v2,r2) c= Ball u,r
for a being set st a in (Ball v1,r1) \/ (Ball v2,r2) holds
a in Ball u,r

proof

let a be set ; :: thesis: ( a in (Ball v1,r1) \/ (Ball v2,r2) implies a in Ball u,r )
assume A1: a in (Ball v1,r1) \/ (Ball v2,r2) ; :: thesis: a in Ball u,r
then reconsider a = a as Point of V ;

now

per cases ( a in Ball v1,r1 or a in Ball v2,r2 ) by A1, XBOOLE_0:def 3;

case a in Ball v1,r1 ; :: thesis: a in Ball u,r

then A2: dist u,a < r1 by BHSP_2:41;
( r1 <= abs r1 & 0 <= abs r2 ) by ABSVALUE:11, COMPLEX1:132;
then A3: r1 + 0 <= (abs r1) + (abs r2) by XREAL_1:9;
0 <= dist v1,v2 by BHSP_1:44;
then r1 + 0 <= ((abs r1) + (abs r2)) + (dist v1,v2) by A3, XREAL_1:9;
then (dist u,a) - r < r1 - r1 by A2, XREAL_1:16;
then ((dist u,a) - r) + r < 0 + r by XREAL_1:10;
hence a in Ball u,r by BHSP_2:41; :: thesis: verum

end;

case a in Ball v2,r2 ; :: thesis: a in Ball u,r

then dist v2,a < r2 by BHSP_2:41;
then (dist v2,a) - (abs r2) < r2 - r2 by ABSVALUE:11, XREAL_1:16;
then ((dist v2,a) - (abs r2)) + (abs r2) < 0 + (abs r2) by XREAL_1:10;
then (dist u,a) - (abs r2) < ((dist v1,v2) + (dist v2,a)) - (dist v2,a) by BHSP_1:42, XREAL_1:17;
then ((dist u,a) - (abs r2)) - (abs r1) < (dist v1,v2) - 0 by COMPLEX1:132, XREAL_1:16;
then ((dist u,a) - ((abs r1) + (abs r2))) + ((abs r1) + (abs r2)) < ((abs r1) + (abs r2)) + (dist v1,v2) by XREAL_1:10;
hence a in Ball u,r by BHSP_2:41; :: thesis: verum

end;

end;

end;

hence a in Ball u,r ; :: thesis: verum

end;

hence (Ball v1,r1) \/ (Ball v2,r2) c= Ball u,r by TARSKI:def 3; :: thesis: verum