reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem Th23:
  rng f = dom g & g*f = f implies g = id dom g
proof
  assume that
A1: rng f = dom g and
A2: g*f = f;
  set X = dom g;
  x in X implies g.x = x
  proof
    assume x in X;
    then ex y being object st y in dom f & f.y = x by A1,Def3;
    hence thesis by A2,Th13;
  end;
  hence thesis by Th17;
end;
