 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 Th69:
  the_unity_wrt the multF of GPFuncs X = id X
proof
  reconsider g = id X as PartFunc of X,X;
  set op = op(GPFuncs X);
A1: carr(GPFuncs X) = PFuncs(X,X) by Def37;
  then reconsider f = g as Element of GPFuncs X by PARTFUN1:45;
  now
    let h be Element of GPFuncs X;
    reconsider j = h as PartFunc of X,X by A1,PARTFUN1:46;
    thus op.(f,h) = f[*]h .= j(*)g by Def37
      .= h by PARTFUN1:6;
    thus op.(h,f) = h[*]f .= g(*)j by Def37
      .= h by PARTFUN1:7;
  end;
  then f is_a_unity_wrt op by BINOP_1:3;
  hence thesis by BINOP_1:def 8;
end;
