 reserve a for non empty set;
 reserve b, x, o for object;

theorem Th10:
    for A be AbGroup, X be non empty set, f be Function of A,X,
    a,b be Element of X st f is bijective holds
    f.((the addF of A).(f".a,f".b)) is Element of X
    proof
      let A be AbGroup, X be non empty set, f be Function of A,X;
      let a,b be Element of X;
      assume
A1:   f is bijective; then
A2:   rng f = X by FUNCT_2:def 3;
      dom f = [#]A by FUNCT_2:def 1; then
reconsider x = f".a, y = f".b as Element of [#]A by A2,A1,FUNCT_1:32;
      reconsider z = (the addF of A).(x,y) as Element of [#]A;
      f.z in X;
      hence thesis;
    end;
