reserve a,a1,a2,b,c,d for Ordinal,
  n,m,k for Nat,
  x,y,z,t,X,Y,Z for set;
reserve f,g for Function;
reserve A,B,C for array;

theorem
  dom A = (limit-A) \ (base-A)
  proof
    consider a,b such that
A1: dom A = a\b by Def1;
A2: limit-A = sup dom A by Th16;
A3: base-A = inf dom A by Th20;
    thus dom A c= (limit-A) \ (base-A)
    proof let x be object;
      reconsider xx=x as set by TARSKI:1;
assume x in dom A; then
A4:   base-A c= xx & x in limit-A by A2,A3,ORDINAL2:14,19; then
A:    x in base-A implies x in xx;
      not xx in xx;
      hence thesis by A,A4,XBOOLE_0:def 5;
    end;
    let x be object; assume
A5: x in (limit-A) \ (base-A); then
    reconsider x as Ordinal;
    ex c st c in dom A & x c= c by A2,A5,ORDINAL2:21; then
    a = limit-A & b = base-A by A1,A2,A3,Th6;
    hence thesis by A1,A5;
  end;
