
theorem
  for G being Group for h being Element of Subgroups G for F being non
  empty Subset of Subgroups G st F = { h } holds meet F = h
proof
  let G be Group;
  let h be Element of Subgroups G;
  let F be non empty Subset of Subgroups G such that
A1: F = { h };
  reconsider H = h as strict Subgroup of G by GROUP_3:def 1;
  h in Subgroups G;
  then h in dom carr G by FUNCT_2:def 1;
  then meet Im(carr G,h) = meet {carr G.h} by FUNCT_1:59;
  then
A2: meet Im(carr G,h) = carr G.h by SETFAM_1:10;
  the carrier of meet F = meet (carr G.:F) by Def2;
  then the carrier of meet F = the carrier of H by A1,A2,Def1;
  hence thesis by GROUP_2:59;
end;
