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
  Y c= field R & R is well-ordering implies R,R |_2 Y are_isomorphic or
  ex a st a in field R & R |_2 (R-Seg(a)),R |_2 Y are_isomorphic
proof
  assume that
A1: Y c= field R and
A2: R is well-ordering;
A3: now
    given a such that
A4: a in field(R |_2 Y) and
A5: R,(R |_2 Y) |_2 ((R |_2 Y)-Seg(a)) are_isomorphic;
    consider F such that
A6: F is_isomorphism_of R,(R |_2 Y) |_2 ((R |_2 Y)-Seg(a)) by A5;
A7: now
      let c,b;
      assume
A8:   [c,b] in R & c <> b;
      then [F.c,F.b] in (R |_2 Y) |_2 ((R |_2 Y)-Seg(a)) by A6;
      then [F.c,F.b] in R |_2 Y by XBOOLE_0:def 4;
      hence [F.c,F.b] in R & F.c <> F.b by A6,A8,Th36,XBOOLE_0:def 4;
    end;
A9: field(R |_2 Y) = Y by A1,A2,Th31;
    field((R |_2 Y) |_2 ((R |_2 Y)-Seg(a))) = (R |_2 Y)-Seg(a) by A2,Th25,Th32;
    then
A10: rng F = (R |_2 Y)-Seg(a) by A6;
A11: dom F = field R by A6;
    then
A12: F.a in rng F by A1,A4,A9,FUNCT_1:def 3;
    then
A13: F.a <> a by A10,Th1;
    [F.a,a] in R |_2 Y by A10,A12,Th1;
    then
A14: [F.a,a] in R by XBOOLE_0:def 4;
    (R |_2 Y)-Seg(a) c= Y by A9,Th9;
    then rng F c= field R by A1,A10;
    then [a,F.a] in R by A1,A2,A4,A9,A11,A7,Th35;
    hence contradiction by A13,A14,A2,Lm3;
  end;
  R |_2 Y is well-ordering by A2,Th25;
  hence thesis by A2,A3,Th52;
end;
