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

theorem Tsym:
   for M,N be LeftMod of R holds M ~= N implies N ~= M
   proof
     let M,N be LeftMod of R;
     assume M ~= N; then
     consider f be Homomorphism of R,M,N such that
A2:  f is one-to-one onto;
     reconsider g = f" as Function of N,M by A2,FUNCT_2:25;
A3:  g is Homomorphism of R,N,M by A2,Th9;
     g*f = id(dom f) by A2,FUNCT_1:39 .= id M by FUNCT_2:def 1; then
     g is onto by FUNCT_2:23;
     hence thesis by A3,A2;
   end;
