 reserve x,y,X,Y for set;
reserve G for non empty multMagma,
  D for set,
  a,b,c,r,l for Element of G;
reserve M for non empty multLoopStr;
reserve H for non empty SubStr of G,
  N for non empty MonoidalSubStr of G;

theorem Th78:
  the_unity_wrt the multF of GPerms X = id X & 1_GPerms X = id X
proof
  reconsider i = id X as Element of GPerms X by Th77;
  now
    let a be Element of GPerms X;
    reconsider f = a as Permutation of X by Th77;
    a[*]i = i(*)a by Th70;
    then
A1: op(GPerms X).(a,i) = i(*)f;
    i[*]a = a(*)i by Th70;
    hence op(GPerms X).(i,a) = a & op(GPerms X).(a,i) = a by A1,FUNCT_2:17;
  end;
  then i is_a_unity_wrt op(GPerms X) by BINOP_1:3;
  hence the_unity_wrt op(GPerms X) = id X by BINOP_1:def 8;
  hence thesis by GROUP_1:22;
end;
