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 Lm3:
  Perm(P,p) is with_fixpoint & Perm(P,q) is without_fixpoints implies
  Perm(P,p=>q) is without_fixpoints
  proof
    given x1 being object such that
A1: x1 is_a_fixpoint_of Perm(P,p);
    reconsider xx1=x1 as set by TARSKI:1;
    assume
A2: for x2 being object holds not x2 is_a_fixpoint_of Perm(P,q);
    let x be object;
    assume
A3: x is_a_fixpoint_of Perm(P,p=>q);
A4: x in dom Perm(P,p=>q) by A3,ABIAN:def 3;
    SetVal(V,p=>q) = Funcs(SetVal(V,p),SetVal(V,q)) by HILBERT3:32;
    then consider f being Function such that
A5: x = f and
    dom f = SetVal(V,p) & rng f c= SetVal(V,q) by A4,FUNCT_2:def 2;
    f.xx1 is_a_fixpoint_of Perm(P,q) by A3,A1,A5,HILBERT3:40;
    hence contradiction by A2;
  end;
