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

theorem Th6:
  R is well-ordering implies for Y st Y c= field R & Y <> {} ex a
  st a in Y & for b st b in Y holds [a,b] in R
proof
  assume
A1: R is well-ordering;
  let Y;
  assume that
A2: Y c= field R and
A3: Y <> {};
  consider a such that
A4: a in Y and
A5: R-Seg(a) misses Y by A1,A2,A3,Def2;
  take a;
  thus a in Y by A4;
  let b;
  assume
A6: b in Y;
  then not b in R-Seg(a) by A5,XBOOLE_0:3;
  then a = b or not [b,a] in R by Th1;
  then a <> b implies [a,b] in R by A1,A2,A4,A6,Lm4;
  hence thesis by A1,A2,A4,Lm1;
end;
