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
  con_class a * A = A * con_class a
proof
  thus con_class a * A c= A * con_class a
  proof
    let x be object;
    assume x in con_class a * A;
    then consider b,c such that
A1: x = b * c and
A2: b in con_class a and
A3: c in A;
    reconsider h = x as Element of G by A1;
    b,a are_conjugated by A2,Th81;
    then consider g such that
A4: b = a |^ g;
    h |^ c" = c * ((a |^ g) * c) * c" by A1,A4
      .= c * (((a |^ g) * c) * c") by GROUP_1:def 3
      .= c * (a |^ g) by Th1;
    then
A5: x = (c * (a |^ g)) |^ c by Th25
      .= (c |^ c) * (a |^ g |^ c) by Th23
      .= c * (a |^ g |^ c) by Th20
      .= c * (a |^ (g * c)) by Th24;
    a |^ (g * c) in con_class a by Th82;
    hence thesis by A3,A5;
  end;
  let x be object;
  assume x in A * con_class a;
  then consider b,c such that
A6: x = b * c and
A7: b in A and
A8: c in con_class a;
  reconsider h = x as Element of G by A6;
  c,a are_conjugated by A8,Th81;
  then consider g such that
A9: c = a |^ g;
  h |^ b = (a |^ g) * b by A6,A9,Th1;
  then
A10: x = ((a |^ g) * b) |^ b" by Th25
    .= (a |^ g) |^ b" * (b |^ b") by Th23
    .= a |^ (g * b") * (b |^ b") by Th24
    .= a |^ (g * b") * b by Th1;
  a |^ (g * b") in con_class a by Th82;
  hence thesis by A7,A10;
end;
