reserve F,G for Group;
reserve G1 for Subgroup of G;
reserve Gc for cyclic Group;
reserve H for Subgroup of Gc;
reserve f for Homomorphism of G,Gc;
reserve a,b for Element of G;
reserve g for Element of Gc;
reserve a1 for Element of G1;
reserve k,m,n,p,s for Element of NAT;
reserve i0,i,i1,i2 for Integer;
reserve j,j1 for Element of INT.Group;
reserve x,y,t for set;

theorem Th3:
  a=a1 implies gr {a} = gr {a1}
proof
  reconsider Gr = gr {a1} as Subgroup of G by GROUP_2:56;
  assume
A1: a= a1;
A2: for b holds b in Gr implies b in gr {a}
  proof
    let b;
    assume
A3: b in Gr;
    then b in G1 by GROUP_2:40;
    then reconsider b1=b as Element of G1 by STRUCT_0:def 5;
    consider i such that
A4: b1= a1|^i by A3,GR_CY_1:5;
    b=a|^i by A1,A4,GROUP_4:2;
    hence thesis by GR_CY_1:5;
  end;
  for b holds b in gr {a} implies b in Gr
  proof
    let b;
    assume b in gr {a};
    then consider i such that
A5: b= a|^i by GR_CY_1:5;
    b=a1|^i by A1,A5,GROUP_4:2;
    hence thesis by GR_CY_1:5;
  end;
  hence thesis by A2,GROUP_2:60;
end;
