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 Th50:
  R is well-ordering & F is_isomorphism_of R,S implies for a st a
  in field R ex b st b in field S & R |_2 (R-Seg(a)),S |_2 (S-Seg(b))
  are_isomorphic
proof
  assume that
A1: R is well-ordering and
A2: F is_isomorphism_of R,S;
  let a;
  assume a in field R;
  then consider b such that
A3: b in field S & F.:(R-Seg(a)) = S-Seg(b) by A2,Th49;
  take b;
  R-Seg(a) c= field R by Th9;
  hence thesis by A1,A2,A3,Th48;
end;
