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;
reserve L for Subset of Subgroups G;
reserve N2 for normal Subgroup of G;

theorem Th117:
  H is normal Subgroup of G iff for a holds a * H = H * a
proof
  thus H is normal Subgroup of G implies for a holds a * H = H * a
  proof
    assume
A1: H is normal Subgroup of G;
    let a;
A2: carr(H |^ a) = a" * H * a by Th59;
    carr(H |^ a) = the carrier of the multMagma of H by A1,Def13
      .= carr H;
    hence a * H = a * (a" * H) * a by A2,GROUP_2:33
      .= a * a" * H * a by GROUP_2:105
      .= 1_G * H * a by GROUP_1:def 5
      .= H * a by GROUP_2:37;
  end;
  assume
A3: for a holds a * H = H * a;
  H is normal
  proof
    let a;
    the carrier of H |^ a = a" * H * a by Th59
      .= H * a" * a by A3
      .= H * (a" * a) by GROUP_2:107
      .= H * 1_G by GROUP_1:def 5
      .= the carrier of H by GROUP_2:109;
    hence thesis by GROUP_2:59;
  end;
  hence thesis;
end;
