
theorem Th7:
  for S being unital non empty multMagma, E being non empty
  finite set, x being Element of E, T being LeftOperation of S, E st x
  is_fixed_under T holds the_orbit_of(x,T) = {x}
proof
  let S be unital non empty multMagma;
  let E be non empty finite set;
  let x be Element of E;
  let T be LeftOperation of S, E;
  set X=the_orbit_of(x,T);
  assume
A1: x is_fixed_under T;
  now
    assume X<>{x};
    then
A2: not (for x9 being object holds x9 in X iff x9=x) by TARSKI:def 1;
    ex x9 being object st x9 in X by XBOOLE_0:def 1;
    then consider x99 be set such that
A3: x99<>x and
A4: x99 in X by A2;
    consider y9 be Element of E such that
A5: x99=y9 and
A6: x,y9 are_conjugated_under T by A4;
    ex s being Element of S st y9 = (T^s).x by A6;
    hence contradiction by A1,A3,A5;
  end;
  hence thesis;
end;
