
theorem Th5:
  for x being object holds product <*{x}*> = {<*x*>}
  proof
    let x be object;
    the set of all <*y*> where y is Element of {x} = {<*x*>}
    proof
      set SA = the set of all <*y*> where y is Element of {x};
A1:   for t be object st t in SA holds t in {<*x*>}
      proof
        let t be object;
        assume t in the set of all <*y*> where y is Element of {x};
        then consider y0 be Element of {x} such that
A2:     t = <*y0*>;
        t = <*x*> by A2,TARSKI:def 1;
        hence thesis by TARSKI:def 1;
      end;
      for t be object st t in {<*x*>} holds t in SA
      proof
        let t be object;
        assume t in {<*x*>};
        then t = <*x*> & x is Element of {x} by TARSKI:def 1;
        hence thesis;
      end;
      then SA c= {<*x*>} & {<*x*>} c= SA by A1;
      hence thesis;
    end;
    hence thesis by SRINGS_4:24;
  end;
