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 Th24:
  R is well_founded implies R |_2 X is well_founded
proof
  assume
A1: for Y st Y c= field R & Y <> {} ex a st a in Y & R-Seg(a) misses Y;
A2: field(R |_2 X) c= field R by Th13;
  let Y;
  assume Y c= field(R |_2 X) & Y <> {};
  then consider a such that
A3: a in Y and
A4: R-Seg(a) misses Y by A1,A2,XBOOLE_1:1;
  take a;
  thus a in Y by A3;
  assume not thesis;
  then
A5: ex b being object st b in (R |_2 X)-Seg(a) & b in Y by XBOOLE_0:3;
  (R |_2 X)-Seg(a) c= R-Seg(a) by Th14;
  hence contradiction by A4,A5,XBOOLE_0:3;
end;
