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 Th4:
  for f,g being Function st Y c= rng(g*f) & g is one-to-one holds g "Y c= rng f
proof
  let f,g be Function such that
A1: Y c= rng(g*f) and
A2: g is one-to-one;
  let y be object;
  assume
A3: y in g"Y;
  then
A4: y in dom g by FUNCT_1:def 7;
  g.y in Y by A3,FUNCT_1:def 7;
  then consider x being object such that
A5: x in dom(g*f) and
A6: g.y = (g*f).x by A1,FUNCT_1:def 3;
A7: x in dom f by A5,FUNCT_1:11;
  g.y = g.(f.x) & f.x in dom g by A5,A6,FUNCT_1:11,12;
  then y = f.x by A2,A4;
  hence thesis by A7,FUNCT_1:def 3;
end;
