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 Th20:
  base-f = inf dom f
  proof set A = f;
    set b = inf dom A;
A1: A is b-based by Th12;
    per cases;
    suppose ex a st a in dom A;
      hence thesis by A1,Def4;
    end;
    suppose
A2:   not ex a st a in dom A;
      set x = the Element of On dom A;
      now
        assume On dom A <> {}; then
        x in dom A by ORDINAL1:def 9;
        hence contradiction by A2;
      end; then
      b = 0 by SETFAM_1:def 1;
      hence thesis by A2,Def4;
    end;
  end;
