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 Th13:
  Gc = gr {g} & g in H implies the multMagma of Gc = the multMagma of H
proof
  assume that
A1: Gc=gr{g} and
A2: g in H;
  reconsider g9=g as Element of H by A2,STRUCT_0:def 5;
  gr {g9} is Subgroup of H;
  then gr {g} is Subgroup of H by Th3;
  hence thesis by A1,GROUP_2:55;
end;
