
theorem
  for G being Group for F being non empty Subset of Subgroups G holds
  (1).G in F implies meet F = (1).G
proof
  let G be Group;
  let F be non empty Subset of Subgroups G;
  assume
A1: (1).G in F;
  reconsider 1G = (1).G as Element of Subgroups G by GROUP_3:def 1;
  carr G.1G = the carrier of (1).G by Def1;
  then the carrier of (1).G in carr G.:F by A1,FUNCT_2:35;
  then {1_G} in carr G.:F by GROUP_2:def 7;
  then meet (carr G.:F) c= {1_G} by SETFAM_1:3;
  then
A2: the carrier of meet F c= {1_G} by Def2;
  (1).G is Subgroup of meet F by GROUP_2:65;
  then the carrier of (1).G c= the carrier of meet F by GROUP_2:def 5;
  then {1_G} c= the carrier of meet F by GROUP_2:def 7;
  then the carrier of meet F = {1_G} by A2;
  hence thesis by GROUP_2:def 7;
end;
