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

theorem Th11:
   for A be AbGroup, X be non empty set, f be Function of A,X,
   a,b,c be Element of X st f is bijective holds
   addemb(f,a,addemb(f,b,c)) = addemb(f,addemb(f,a,b),c)
   proof
     let A be AbGroup, X be non empty set, f be Function of A,X;
     let a,b,c be Element of X;
     assume
A1:  f is bijective; then
A2:  rng f = X by FUNCT_2:def 3;
A3:  dom f = [#]A by FUNCT_2:def 1;
reconsider x=f".a, y=f".b, z=f".c as Element of [#]A by A3,A2,A1,FUNCT_1:32;
A4:  addemb(f,a,b) = f.((the addF of A).(f".a,f".b)) by A1,Def8;
     addemb(f,b,c) = f.((the addF of A).(f".b,f".c)) by A1,Def8; then
 reconsider ab1 = f.((the addF of A).(f".a,f".b)),
     bc1 = f.((the addF of A).(f".b,f".c)) as Element of X by A4;
     addemb(f,a,addemb(f,b,c)) = addemb(f,a,bc1) by A1,Def8
     .= f.((the addF of A).(f".a,f".bc1)) by A1,Def8
     .= f.(x+(y+z)) by A1,A3,FUNCT_1:34
     .= f.((x+y)+z) by RLVECT_1:def 3
     .= f.((the addF of A).(f".ab1,f".c)) by A1,A3,FUNCT_1:34
     .= addemb(f,ab1,c) by A1,Def8
     .= addemb(f,addemb(f,a,b),c) by A1,Def8;
     hence thesis;
   end;
