reserve
   a,b,c,x,y,z,A,B,C,X,Y for set,
   f,g for Function,
   V for SetValuation,
   P for Permutation of V,
   p,q,r,s for Element of HP-WFF,
   n for Element of NAT;

theorem Th14:
  x is_a_fixpoint_of Perm(P,q) implies
  SetVal(V,p) --> x is_a_fixpoint_of Perm(P,p=>q)
  proof
    assume
A1: x is_a_fixpoint_of Perm(P,q);
    set F = SetVal(V,p) --> x;
A2: dom Perm(P,p=>q) = SetVal(V,p=>q) by FUNCT_2:def 1;
A3: SetVal(V,p=>q) = Funcs(SetVal(V,p),SetVal(V,q)) by HILBERT3:32;
    x in dom Perm(P,q) by A1,ABIAN:def 3;
    then
A4: F is Function of SetVal(V,p),SetVal(V,q) by FUNCOP_1:46;
    hence
A5: F in dom Perm(P,p=>q) by A2,A3,FUNCT_2:8;
A6: Perm(P,p=>q) = Perm(P,p)=>Perm(P,q) by HILBERT3:36;
    Perm(P,p=>q).F in SetVal(V,p=>q) by A5,FUNCT_2:5;
    then
A7: dom (Perm(P,p=>q).F) = SetVal(V,p) by A3,FUNCT_2:92;
    now
      let z be object such that
A8:   z in dom (Perm(P,p=>q).F);
      (Perm(P,p)=>Perm(P,q)).F =
      Perm(P,q)*F*Perm(P,p)" by A4,HILBERT3:def 1;
      hence Perm(P,p=>q).F.z = (Perm(P,q)*F).((Perm(P,p)").z)
      by A6,A8,FUNCT_1:12
      .= Perm(P,q).(F.(Perm(P,p)".z)) by A7,A8,FUNCT_2:5,15
      .= Perm(P,q).x by A7,A8,FUNCT_2:5,FUNCOP_1:7
      .= x by A1,ABIAN:def 3;
    end;
    hence thesis by A7,FUNCOP_1:11;
  end;
