theorem Th31:
  (k+1)-eq_states_partition tfsm <> k-eq_states_partition tfsm
  implies for i st i <= k holds (i+1)-eq_states_partition tfsm <> i
  -eq_states_partition tfsm
proof
  assume
A1: (k+1)-eq_states_partition tfsm <> k-eq_states_partition tfsm;
  let i be Nat such that
A2: i <= k;
A3: ex e being Nat st k+1 = i+e by A2,NAT_1:10,12;
  assume
A4: (i+1)-eq_states_partition tfsm = i-eq_states_partition tfsm;
  ex d being Nat st k = i+d by A2,NAT_1:10;
  then k-eq_states_partition tfsm = i-eq_states_partition tfsm by A4,Th29;
  hence contradiction by A1,A4,A3,Th29;
end;
