reserve x,y for set,
  k,n for Nat,
  i for Integer,
  G for Group,
  a,b,c ,d,e for Element of G,
  A,B,C,D for Subset of G,
  H,H1,H2,H3,H4 for Subgroup of G ,
  N1,N2 for normal Subgroup of G,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem
  a in center G iff con_class a = {a}
proof
  thus a in center G implies con_class a = {a}
  proof
    assume
A1: a in center G;
    thus con_class a c= {a}
    proof
      let x be object;
      assume x in con_class a;
      then consider b such that
A2:   b = x and
A3:   a,b are_conjugated by GROUP_3:80;
      consider c such that
A4:   a = b |^ c by A3;
      a = c" * (b * c) by A4,GROUP_1:def 3;
      then c * a = b * c by GROUP_1:13;
      then a * c = b * c by A1,Th77;
      then a = b by GROUP_1:6;
      hence thesis by A2,TARSKI:def 1;
    end;
    a in con_class a by GROUP_3:83;
    hence thesis by ZFMISC_1:31;
  end;
  assume
A5: con_class a = {a};
  now
    let b;
    a |^ b in con_class a by GROUP_3:82;
    then a |^ b = a by A5,TARSKI:def 1;
    then b" * (a * b) = a by GROUP_1:def 3;
    hence a * b = b * a by GROUP_1:13;
  end;
  hence thesis by Th77;
end;
