reserve X,Y,Z for set,
  a,b,c,d,x,y,z,u for object,
  R for Relation,
  A,B,C for Ordinal;
reserve H for Function;

theorem Th3:
  A in B implies A = (RelIncl B)-Seg(A)
proof
  assume
A1: A in B;
  thus for a being object holds a in A implies a in (RelIncl B)-Seg(A)
  proof let a be object;
    assume
A2: a in A;
    then reconsider C = a as Ordinal;
    reconsider a as set by TARSKI:1;
    not a in a; then
A3: a <> A by A2;
A4: A c= B by A1,ORDINAL1:def 2;
    C c= A by A2,ORDINAL1:def 2;
    then [C,A] in RelIncl B by A1,A2,A4,Def1;
    hence thesis by A3,WELLORD1:1;
  end;
  let a be object;
  assume
A5: a in (RelIncl B)-Seg(A);
  then
A6: a <> A by WELLORD1:1;
A7: [a,A] in RelIncl B by A5,WELLORD1:1;
  then
A8: a in field RelIncl B by RELAT_1:15;
A9: field RelIncl B = B by Def1;
  then reconsider C = a as Ordinal by A8;
  C c= A by A1,A7,A8,A9,Def1;
  then C c< A by A6;
  hence thesis by ORDINAL1:11;
end;
