
theorem
  for G1, G2 being Group for f being Homomorphism of G1, G2 holds (
  FuncLatt f).(1).G1 = (1).G2
proof
  let G1, G2 be Group;
  let f be Homomorphism of G1, G2;
  consider A being Subset of G2 such that
A1: A = f.:the carrier of (1).G1;
A2: A = f.:{1_G1} by A1,GROUP_2:def 7;
A3: 1_G1 in {1_G1} & 1_G2 = f.1_G1 by GROUP_6:31,TARSKI:def 1;
  for x being object holds x in A iff x = 1_G2
  proof
    let x be object;
    thus x in A implies x = 1_G2
    proof
      assume
A4:   x in A;
      then reconsider x as Element of G2;
      consider y being Element of G1 such that
A5:   y in {1_G1} and
A6:   x = f.y by A2,A4,FUNCT_2:65;
      y = 1_G1 by A5,TARSKI:def 1;
      hence thesis by A6,GROUP_6:31;
    end;
    thus thesis by A2,A3,FUNCT_2:35;
  end;
  then A = {1_G2} by TARSKI:def 1;
  then gr A = (1).G2 by Th12;
  hence thesis by A1,Def3;
end;
