reserve Q,Q1,Q2 for multLoop;
reserve x,y,z,w,u,v for Element of Q;

theorem Th57:
  L_MAP(x,y) in InnAut Q
proof
  set f = L_MAP(x,y);
  reconsider g = (curry (the multF of Q)).(y * x) as
    Permutation of the carrier of Q by Th30;
  reconsider h = (curry (the multF of Q)).x as
    Permutation of the carrier of Q by Th30;
  reconsider k = (curry (the multF of Q)).(y) as
    Permutation of the carrier of Q by Th30;
  A2: f = g" * (k * h)
  proof
    for u holds (g*f).u = (k*h).u
    proof
      let u;
      (g*f).u = g.(f.u) by FUNCT_2:15
      .= g.(L_map(u,x,y)) by LM1
      .= (y * x) * ((y * x) \ (y * (x * u))) by FUNCT_5:69
      .= k.(x * u) by FUNCT_5:69
      .= k.(h.u) by FUNCT_5:69
      .= (k*h).u by FUNCT_2:15;
      hence thesis;
    end;
    then g"*(k*h) = g"*(g*f) by FUNCT_2:def 8
    .= (g"*g)*f by RELAT_1:36
    .= (id the carrier of Q)*f by FUNCT_2:61
    .= f by FUNCT_2:17;
    hence thesis;
  end;
  g in Mlt ([#] Q) by Th32;
  then A3: g" in Mlt ([#] Q) by Def35;
  h in Mlt ([#] Q) & k in Mlt ([#] Q) by Th32;
  then A4:k * h in Mlt ([#] Q) by Def34;
  f.(1.Q) = L_map(1.Q,x,y) by LM1
  .= 1.Q by Th5;
  hence thesis by Th55,A4,A2,Def34,A3;
end;
