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 a-based segmental iff ex b st dom f = b\a & a c= b
  proof
    thus f is a-based segmental implies ex b st dom f = b\a & a c= b
    proof
      assume
A1:   b in dom f implies a in dom f & a c= b;
      given c,d such that
A2:   dom f = c\d;
      set x = the Element of c\d;
      per cases;
      suppose dom f = {}; then
        a\a = dom f by XBOOLE_1:37;
        hence thesis;
      end;
      suppose
        dom f <> {}; then
        x in dom f by A2; then
        a in dom f by A1; then
A3:     d c= a & a in c by A2,ORDINAL6:5; then
        d in c by ORDINAL1:12; then
        d in dom f by A2,ORDINAL6:5; then
        a c= d by A1; then
        a = d & a c= c by A3,ORDINAL1:def 2;
        hence thesis by A2;
      end;
    end;
    given b such that
A4: dom f = b\a & a c= b;
    thus f is a-based
    proof
      let c; assume
A5:   c in dom f; then
      a c= c & c in b by A4,ORDINAL6:5; then
      a in b by ORDINAL1:12;
      hence thesis by A4,A5,ORDINAL6:5;
    end;
    take b,a; thus thesis by A4;
  end;
