reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem Th47:
  rng delta X c= [:X,X:]
proof
  let y be object;
  assume y in rng delta X;
  then consider x being object such that
A1: x in dom delta X and
A2: y = (delta X).x by FUNCT_1:def 3;
A3: x in X by A1,Def6;
  then y = [x,x] by A2,Def6;
  hence thesis by A3,ZFMISC_1:87;
end;
