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

theorem
  for R st R is well-ordering for a st a in field R holds (for b st b in
field R holds [b,a] in R) or ex b st b in field R & [a,b] in R & for c st c in
  field R & [a,c] in R holds c = a or [b,c] in R
proof
  let R;
  assume
A1: R is well-ordering;
  let a such that
A2: a in field R;
  defpred P[object] means not [$1,a] in R;
  given b such that
A3: b in field R & not [b,a] in R;
  consider Z such that
A4: for c being object holds c in Z iff c in field R & P[c]
from XBOOLE_0:sch 1;
  for b being object holds b in Z implies b in field R by A4;
  then
A5: Z c= field R;
  Z <> {} by A3,A4;
  then consider d such that
A6: d in Z and
A7: for c st c in Z holds [d,c] in R by A1,A5,Th6;
  take d;
  thus
A8: d in field R by A4,A6;
A9: not [d,a] in R by A4,A6;
  then a <> d by A6,A7;
  hence [a,d] in R by A1,A2,A8,A9,Lm4;
  let c;
  assume that
A10: c in field R and
A11: [a,c] in R;
  assume c <> a;
  then not [c,a] in R by A1,A11,Lm3;
  then c in Z by A4,A10;
  hence thesis by A7;
end;
