reserve r,s,t,u for Real;

theorem
  for X being non empty RLSStruct, F being circled-membered
  Subset-Family of X holds union F is circled
proof
  let X be non empty RLSStruct, F be circled-membered Subset-Family of X;
  let r be Real such that
A1: |.r.| <= 1;
  let x be object;
  assume x in r*(union F);
  then consider x9 being Point of X such that
A2: x = r*x9 and
A3: x9 in union F;
  consider Y being set such that
A4: x9 in Y and
A5: Y in F by A3,TARSKI:def 4;
  reconsider Y as Subset of X by A5;
  Y is circled by A5,Def7;
  then
A6: r*Y c= Y by A1;
  r*x9 in r*Y by A4;
  hence thesis by A2,A5,A6,TARSKI:def 4;
end;
