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 Th45:
  rng pr2(X,Y) c= Y
proof
  let y be object;
  assume y in rng pr2(X,Y);
  then consider p being object such that
A1: p in dom pr2(X,Y) and
A2: y = pr2(X,Y).p by FUNCT_1:def 3;
  p in [:X,Y:] by A1,Def5;
  then consider x1,y1 being object such that
A3: x1 in X & y1 in Y and
A4: p = [x1,y1] by ZFMISC_1:def 2;
  y = pr2(X,Y).(x1,y1) by A2,A4;
  hence thesis by A3,Def5;
end;
