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;

theorem
  f is b-limited non empty segmental iff ex a st dom f = b\a & a in b
  proof
    thus f is b-limited non empty segmental implies
    ex a st dom f = b\a & a in b
    proof assume
A1:   b = sup dom f;
      assume
A2:   f is non empty;
      given c,d such that
A3:   dom f = c\d;
      set x = the Element of c\d;
      take a = d;
A4:   b = c by A2,A1,A3,Th6;
      thus dom f = b\a by A2,A1,A3,Th6;
      a c= x & x in b by A2,A3,A4,ORDINAL6:5;
      hence a in b by ORDINAL1:12;
    end;
    given a such that
A5: dom f = b\a & a in b;
    a in dom f by A5,ORDINAL6:5;
    hence b = sup dom f by A5,Th6;
    thus f is non empty by A5,ORDINAL6:5;
    take b,a;
    thus thesis by A5;
  end;
