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

theorem Th58:
  R_MAP(x,y) in InnAut Q
proof
  set f = R_MAP(x,y);
  reconsider g = (curry' (the multF of Q)).(x * y) as
    Permutation of the carrier of Q by Th31;
  reconsider h = (curry' (the multF of Q)).x as
    Permutation of the carrier of Q by Th31;
  reconsider k = (curry' (the multF of Q)).(y) as
    Permutation of the carrier of Q by Th31;
  A2: f = g" * (k * h)
  proof
    for u holds (g*f).u = (k*h).u
    proof
      let u;
      thus (g*f).u = g.(f.u) by FUNCT_2:15
      .= g.(R_map(u,x,y)) by RM1
      .= (((u * x) * y) / (x * y)) * (x * y) by FUNCT_5:70
      .= k.(u * x) by FUNCT_5:70
      .= k.(h.u) by FUNCT_5:70
      .= (k*h).u by FUNCT_2:15;
    end;
    then g"*(k*h) = g"*(g*f) by FUNCT_2:def 8
    .= (g"*g)*f by RELAT_1:36
    .= (id Q)*f by FUNCT_2:61
    .= f by FUNCT_2:17;
    hence thesis;
  end;
  g in Mlt ([#] Q) by Th33;
  then A3: g" in Mlt ([#] Q) by Def35;
  h in Mlt ([#] Q) & k in Mlt ([#] Q) by Th33;
  then A4: k * h in Mlt ([#] Q) by Def34;
  f.(1.Q) = R_map(1.Q,x,y) by RM1
  .= 1.Q by Th6;
  hence thesis by Th55,A4,A2,Def34,A3;
end;
