
theorem
  for G being Group holds FuncLatt id the carrier of G = id the carrier
  of lattice G
proof
  let G be Group;
  set f = id the carrier of G;
A1: for x being object st x in the carrier of lattice G
    holds (FuncLatt f).x = x
  proof
    let x be object;
    assume x in the carrier of lattice G;
    then reconsider x as strict Subgroup of G by GROUP_3:def 1;
A2: the carrier of x c= f.:the carrier of x
    proof
      let z be object;
      assume
A3:   z in the carrier of x;
      the carrier of x c= the carrier of G by GROUP_2:def 5;
      then reconsider z as Element of G by A3;
      f.z = z;
      hence thesis by A3,FUNCT_2:35;
    end;
A4: f.:the carrier of x c= the carrier of x
    proof
      let z be object;
      assume
A5:   z in f.:the carrier of x;
      then reconsider z as Element of G;
      ex Z being Element of G st Z in the carrier of x & z = f. Z by A5,
FUNCT_2:65;
      hence thesis;
    end;
    then reconsider X = the carrier of x as Subset of G by A2,XBOOLE_0:def 10;
    f.:the carrier of x = the carrier of x by A4,A2;
    then (FuncLatt f).x = gr X by Def3;
    hence thesis by Th3;
  end;
  dom FuncLatt f = the carrier of lattice G by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:17;
end;
