reserve k,n,m for Nat,
  A,B,C for Ordinal,
  X for set,
  x,y,z for object;
reserve f,g,h,fx for Function,
  K,M,N for Cardinal,
  phi,psi for
  Ordinal-Sequence;

theorem Th5:
  X c= A implies ex phi st phi = canonical_isomorphism_of(RelIncl
  order_type_of RelIncl X, RelIncl X) & phi is increasing & dom phi =
  order_type_of RelIncl X & rng phi = X
proof
  set R = RelIncl X;
  set B = order_type_of R;
  set phi = canonical_isomorphism_of (RelIncl B,R);
  assume
A1: X c= A;
  then R is well-ordering by WELLORD2:8;
  then R, RelIncl B are_isomorphic by WELLORD2:def 2;
  then RelIncl B is well-ordering & RelIncl B, R are_isomorphic by WELLORD1:40
,WELLORD2:8;
  then
A2: phi is_isomorphism_of RelIncl B,R by WELLORD1:def 9;
  then
A3: phi is one-to-one by WELLORD1:def 7;
A4: field RelIncl B = B by WELLORD2:def 1;
  then
A5: dom phi = B by A2,WELLORD1:def 7;
A6: field R = X by WELLORD2:def 1;
  then
A7: rng phi = X by A2,WELLORD1:def 7;
  reconsider phi as Sequence by A5,ORDINAL1:def 7;
  reconsider phi as Ordinal-Sequence by A1,A7,ORDINAL2:def 4;
  take phi;
  thus phi = canonical_isomorphism_of (RelIncl order_type_of RelIncl X,
  RelIncl X);
  thus phi is increasing
  proof
    let a,b be Ordinal;
    assume that
A8: a in b and
A9: b in dom phi;
A10: a in dom phi by A8,A9,ORDINAL1:10;
    reconsider a9 = phi.a, b9 = phi.b as Ordinal;
A11: b9 in X by A7,A9,FUNCT_1:def 3;
    a c= b by A8,ORDINAL1:def 2;
    then [a,b] in RelIncl B by A5,A9,A10,WELLORD2:def 1;
    then
A12: [a9,b9] in R by A2,WELLORD1:def 7;
    a9 in X by A7,A10,FUNCT_1:def 3;
    then
A13: a9 c= b9 by A12,A11,WELLORD2:def 1;
    a <> b by A8;
    then a9 <> b9 by A3,A9,A10;
    then a9 c< b9 by A13;
    hence thesis by ORDINAL1:11;
  end;
  thus thesis by A2,A4,A6,WELLORD1:def 7;
end;
