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
  Y = rng f & dom g = Y & dom h = Y & g*f = h*f implies g = h
proof
  assume that
A1: Y = rng f and
A2: dom g = Y & dom h = Y and
A3: g*f = h*f;
  y in Y implies g.y = h.y
  proof
    assume y in Y;
    then consider x being object such that
A4: x in dom f & y = f.x by A1,Def3;
    (g*f).x = g.y by A4,Th13;
    hence thesis by A3,A4,Th13;
  end;
  hence thesis by A2,Th2;
end;
