reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;

theorem Th100:
  con_class{a} = {{b} : b in con_class a}
proof
  set A = {{b} : b in con_class a};
  thus con_class{a} c= A
  proof
    let x be object;
    assume x in con_class{a};
    then consider B such that
A1: x = B and
A2: {a},B are_conjugated;
    consider b such that
A3: {a} |^ b = B by A2,Th88;
    a,a |^ b are_conjugated by Th74;
    then
A4: a |^ b in con_class a by Th81;
    B = {a |^ b} by A3,Th37;
    hence thesis by A1,A4;
  end;
  let x be object;
  assume x in A;
  then consider b such that
A5: x = {b} and
A6: b in con_class a;
  b,a are_conjugated by A6,Th81;
  then {b},{a} are_conjugated by Th92;
  hence thesis by A5;
end;
