reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;

theorem
  for x being set holds x is_a_fixpoint_of {[x,x]}
  proof
    let x be set;
A1: [x,x] in {[x,x]} by TARSKI:def 1;
    dom {[x,x]} = {x} by RELAT_1:9;
    hence x in dom {[x,x]} by TARSKI:def 1;
    hence x = {[x,x]}.x by A1,FUNCT_1:def 2;
  end;
