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 Th7:
  for R st R is well-ordering ex A st R,RelIncl A are_isomorphic
proof
  let R such that
A1: R is well-ordering;
  defpred P[object] means ex A st R |_2 (R-Seg($1)),RelIncl A are_isomorphic;
  consider Z such that
A2: for a being object holds a in Z iff a in field R & P[a]
from XBOOLE_0:sch 1;
  now
    let a be object such that
A3: a in field R and
A4: R-Seg(a) c= Z;
    set P = R |_2 (R-Seg(a));
    now
      let b be object;
      assume
A5:   b in field P;
      then
A6:   b in R-Seg(a) by WELLORD1:12;
      then
A7:   [b,a] in R by WELLORD1:1;
      b in field R by A5,WELLORD1:12;
      then
A8:   R-Seg(b) c= R-Seg(a) by A1,A3,A7,WELLORD1:29;
      consider A such that
A9:   R |_2 (R-Seg(b)),RelIncl A are_isomorphic by A2,A4,A6;
      take A;
      P-Seg(b) = R-Seg(b) by A1,A5,WELLORD1:12,27;
      hence P |_2 (P-Seg(b)),RelIncl A are_isomorphic by A9,A8,WELLORD1:22;
    end;
    then ex A st P,RelIncl A are_isomorphic by A1,Th6,WELLORD1:25;
    hence a in Z by A2,A3;
  end;
  then field R c= Z by A1,WELLORD1:33;
  then
  for a being object st a in field R
ex A st R |_2 (R-Seg(a)),RelIncl A are_isomorphic
  by A2;
  hence thesis by A1,Th6;
end;
