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;

theorem Th67:
  (1).G |^ a = (1).G
proof
A1: the carrier of (1).G = {1_G} by GROUP_2:def 7;
  the carrier of (1).G |^ a = (carr (1).G) |^ a by Def6
    .= {(1_G) |^ a} by A1,Th37
    .= {1_G} by Th17;
  hence thesis by GROUP_2:def 7;
end;
