theorem Th12:
  f is_a_proof_wrt_IPC X implies Effect_IPC(f) in CnIPC(X)
proof
  assume
A1: f is_a_proof_wrt_IPC X; then
A2: 1 <= len f by Th5; then
A3: (f.(len f))`1 in CnIPC(X) by A1,Th11;
  f <> {} by A2;
  hence thesis by A3,Def5;
end;
