
theorem Th15:
  for V being non empty RLSStruct, F being Subset-Family of V st (
  for M being Subset of V st M in F holds M is convex) holds meet F is convex
proof
  let V be non empty RLSStruct;
  let F be Subset-Family of V;
  assume
A1: for M being Subset of V st M in F holds M is convex;
  per cases;
  suppose
    F = {};
    then meet F = {} by SETFAM_1:def 1;
    hence thesis;
  end;
  suppose
A2: F <> {};
    meet F is convex
    proof
      let u,v be VECTOR of V;
      let r be Real;
      assume that
A3:   0 < r & r < 1 and
A4:   u in meet F and
A5:   v in meet F;
      for M being set st M in F holds r*u + (1-r)*v in M
      proof
        let M be set;
        assume
A6:     M in F;
        then reconsider M as Subset of V;
A7:     v in M by A5,A6,SETFAM_1:def 1;
        M is convex & u in M by A1,A4,A6,SETFAM_1:def 1;
        hence thesis by A3,A7;
      end;
      hence thesis by A2,SETFAM_1:def 1;
    end;
    hence thesis;
  end;
end;
