reserve n for Element of NAT,
  p,q,r,s for Element of HP-WFF;
reserve V for SetValuation;
reserve P for Permutation of V;

theorem Th38:
  for f,g being Function of SetVal(V,p), SetVal(V,q) st f = Perm(P
  ,p => q).g holds Perm(P,q)*g = f*Perm(P,p)
proof
  let f,g be Function of SetVal(V,p), SetVal(V,q) such that
A1: f = Perm(P,p => q).g;
  thus Perm(P,q)*g = Perm(P,q)*g*(id SetVal(V,p)) by FUNCT_2:17
    .= Perm(P,q)*g*(Perm(P,p)"*Perm(P,p)) by FUNCT_2:61
    .= Perm(P,q)*g*Perm(P,p)"*Perm(P,p) by RELAT_1:36
    .= f*Perm(P,p) by A1,Th36;
end;
