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 Th80:
  x in con_class a iff ex b st b = x & a,b are_conjugated
proof
  thus x in con_class a implies ex b st b = x & a,b are_conjugated
  proof
    assume x in con_class a;
    then x in {b : a,b are_conjugated} by Th79;
    hence thesis;
  end;
  given b such that
A1: b = x & a,b are_conjugated;
  x in {c : a,c are_conjugated} by A1;
  hence thesis by Th79;
end;
