 reserve i,j,n,k,l for Nat;
 reserve T,S,X,Y,Z for Subset of MC-wff;
 reserve p,q,r,t,F,H,G for Element of MC-wff;
 reserve s,U,V for MC-formula;
reserve f,g for FinSequence of [:MC-wff,Proof_Step_Kinds_IPC:];

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;
