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;
