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 Th31:
  R is well-ordering & X c= field R implies field(R |_2 X) = X
proof
  assume that
A1: R is well-ordering and
A2: X c= field R;
  thus field(R |_2 X) c= X by Th13;
  let x be object;
  assume
A3: x in X;
  then
A4: [x,x] in [:X,X:] by ZFMISC_1:87;
  [x,x] in R by A1,A2,A3,Lm1;
  then [x,x] in R |_2 X by A4,XBOOLE_0:def 4;
  hence thesis by RELAT_1:15;
end;
