reserve i,j,k,x for object;

theorem
  for C being with_units non empty AltCatStr holds C is
  pseudo-discrete iff for o being Object of C holds <^o,o^> = { idm o }
proof
  let C be with_units non empty AltCatStr;
  hereby
    assume
A1: C is pseudo-discrete;
    let o be Object of C;
    now
      let j be object;
      hereby
        idm o in <^o,o^> & <^o,o^> is trivial by A1,Th13;
        then consider i being object such that
A2:     <^o,o^> = { i } by ZFMISC_1:131;
        assume j in <^o,o^>;
        then j = i by A2,TARSKI:def 1;
        hence j = idm o by A2,TARSKI:def 1;
      end;
      thus j = idm o implies j in <^o,o^> by Th13;
    end;
    hence <^o,o^> = { idm o } by TARSKI:def 1;
  end;
  assume
A3: for o be Object of C holds <^o,o^> = { idm o };
  let o be Object of C;
  <^o,o^> = { idm o } by A3;
  hence thesis;
end;
