reserve a,b,c,d,x,y,z for object, X,Y,Z for set;
reserve R,S,T for Relation;
reserve F,G for Function;

theorem Th33:
  R is well-ordering implies for Z st for a st a in field R & R
  -Seg(a) c= Z holds a in Z holds field R c= Z
proof
  assume
A1: R is well-ordering;
  let Z such that
A2: for a st a in field R & R-Seg(a) c= Z holds a in Z;
A3: now
    let a such that
A4: a in field R and
A5: for b st [b,a] in R & a <> b holds b in Z;
    now
      let b be object;
      assume b in R-Seg(a);
      then [b,a] in R & b <> a by Th1;
      hence b in Z by A5;
    end;
    then R-Seg(a) c= Z;
    hence a in Z by A2,A4;
  end;
  given a being object such that
A6: a in field R & not a in Z;
  field R \ Z <> {} by A6,XBOOLE_0:def 5;
  then consider a such that
A7: a in field R \ Z and
A8: for b st b in field R \ Z holds [a,b] in R by A1,Th6;
  not a in Z by A7,XBOOLE_0:def 5;
  then consider b such that
A9: [b,a] in R and
A10: b <> a and
A11: not b in Z by A3,A7;
  b in field R by A9,RELAT_1:15;
  then b in field R \ Z by A11,XBOOLE_0:def 5;
  then [a,b] in R by A8;
  hence contradiction by A1,A9,A10,Lm3;
end;
