reserve r,s,t,u for Real;

theorem
  for X being non empty RLSStruct, F being circled-membered
  Subset-Family of X holds meet 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*(meet F);
  then consider x9 being Point of X such that
A2: x = r*x9 and
A3: x9 in meet F;
A4: now
    let Y be set;
    assume
A5: Y in F;
    then reconsider Y9 = Y as Subset of X;
    x9 in Y by A3,A5,SETFAM_1:def 1;
    then
A6: r*x9 in r*Y9;
    Y9 is circled by A5,Def7;
    then r*Y9 c= Y9 by A1;
    hence x in Y by A2,A6;
  end;
  F <> {} by A3,SETFAM_1:def 1;
  hence thesis by A4,SETFAM_1:def 1;
end;
