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

theorem
  R well_orders X implies for Y st Y c= X & Y <> {} ex a st a in Y & for
  b st b in Y holds [a,b] in R
proof
  assume
A1: R well_orders X;
  then
A2: R is_reflexive_in X;
A3: R is_connected_in X by A1;
  let Y;
  assume that
A4: Y c= X and
A5: Y <> {};
  R is_well_founded_in X by A1;
  then consider a such that
A6: a in Y and
A7: R-Seg(a) misses Y by A4,A5;
  take a;
  thus a in Y by A6;
  let b;
  assume
A8: b in Y;
  then not b in R-Seg(a) by A7,XBOOLE_0:3;
  then a = b or not [b,a] in R by Th1;
  then a <> b implies [a,b] in R by A3,A4,A6,A8,RELAT_2:def 6;
  hence thesis by A2,A4,A6,RELAT_2:def 1;
end;
