reserve M,N for AbGroup;
 reserve R for Ring;
 reserve r for Element of R;

theorem
   for L,M,N be LeftMod of R holds L ~= M & M ~= N implies L ~= N
   proof
     let L,M,N be LeftMod of R;
     assume
A1:  L ~= M & M ~= N; then
     consider f be Homomorphism of R,L,M such that
A2:  f is one-to-one onto;
     consider g be Homomorphism of R,M,N such that
A3:  g is one-to-one onto by A1;
     reconsider G = g*f as Function of L,N;
     for x,y be Element of L holds G.(x+y) = G.x + G.y
     proof
       let x,y be Element of L;
A4:    G.x = g.(f.x) by FUNCT_2:15;
A5:    G.y = g.(f.y) by FUNCT_2:15;
       G.(x+y) = g.(f.(x+y)) by FUNCT_2:15
       .= g.(f.x+f.y) by Def10,VECTSP_1:def 20
       .= G.x + G.y by A4,A5,Def10,VECTSP_1:def 20;
       hence thesis;
     end; then
A6:  G is additive;
     for x be Element of L, a be Element of R holds G.(a*x) = a*G.x
     proof
       let x be Element of L, a be Element of R;
A7:    G.x = g.(f.x) by FUNCT_2:15;
       G.(a*x) = g.(f.(a*x)) by FUNCT_2:15
       .= g.(a*f.x) by Def10,MOD_2:def 2
       .= a*G.x by A7,Def10,MOD_2:def 2;
       hence thesis;
     end; then
     G is homogeneous; then
A8:  G is Homomorphism of R,L,N by A6,Def10;
     G is one-to-one onto by A2,A3,FUNCT_2:27;
     hence thesis by A8;
   end;
