reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;

theorem Th61:
  for f being Permutation of X st P c= X holds f.:(f"P) = P & f"(f .:P) = P
proof
  let f be Permutation of X such that
A1: P c= X;
  dom f = X by Th51;
  then
A2: P c= f"(f.:P) by A1,FUNCT_1:76;
  f"(f.:P) c= P & rng f = X by Def3,FUNCT_1:82;
  hence thesis by A1,A2,FUNCT_1:77;
end;
