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

theorem Th56:
  T_MAP(x) in InnAut Q
proof
  set f = T_MAP(x);
  reconsider g = (curry (the multF of Q)).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 Th31;
  A2: f = g" * h
  proof
    for u holds (g * f).u = h.u
    proof
      let u;
      thus (g * f).u = g.(f.u) by FUNCT_2:15
      .= g.(T_map(u,x)) by TM1
      .= x * (x \ (u * x)) by FUNCT_5:69
      .= h.u by FUNCT_5:70;
    end;
    then g"*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;
  A4: h in Mlt ([#] Q) by Th33;
  f.(1.Q) = T_map(1.Q,x) by TM1
  .= 1.Q by Th5;
  hence thesis by A4,Th55,A2,Def34,A3;
end;
