reserve m,n for Nat;
reserve i,j for Integer;
reserve S for non empty addMagma;
reserve r,r1,r2,s,s1,s2,t,t1,t2 for Element of S;
reserve G for addGroup-like non empty addMagma;
reserve e,h for Element of G;
reserve G for addGroup;
reserve f,g,h for Element of G;
reserve u for UnOp of G;
reserve A for Abelian addGroup;
reserve a,b for Element of A;
reserve x for object;
reserve y,y1,y2,Y,Z for set;
reserve k for Nat;
reserve G for addGroup;
reserve a,g,h for Element of G;
reserve A for Subset of G;
reserve G for non empty addMagma,
  A,B,C for Subset of G;
reserve a,b,g,g1,g2,h,h1,h2 for Element of G;
reserve G for addGroup-like non empty addMagma;
reserve h,g,g1,g2 for Element of G;
reserve A for Subset of G;
reserve H for Subgroup of G;
reserve h,h1,h2 for Element of H;
reserve G,G1,G2,G3 for addGroup;
reserve a,a1,a2,b,b1,b2,g,g1,g2 for Element of G;
reserve A,B for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve h,h1,h2 for Element of H;
reserve x,y,y1,y2 for set;
reserve G for addGroup;
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,RLVECT_1:def 3
      .= c + (a * g) by Th1;
    then
A5: x = (c + (a * g)) * c by ThB25
      .= (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 Th80,Th74;
    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 ThB25
    .= (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 Th80,Th74;
  hence thesis by A7,A10;
end;
