 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 Th16:
  p in CnIPC(X) iff ex f st f is_a_proof_wrt_IPC X & Effect_IPC(f) = p
proof
  thus p in CnIPC(X) implies ex f st f is_a_proof_wrt_IPC X &
  Effect_IPC(f) = p
  proof
    assume p in CnIPC(X);
    then p in {F: ex f st f is_a_proof_wrt_IPC X & Effect_IPC(f) = F}
      by Th15;
    then ex F st F=p & ex f st f is_a_proof_wrt_IPC X & Effect_IPC(f) = F;
    hence thesis;
  end;
    given f such that
A1: f is_a_proof_wrt_IPC X & Effect_IPC(f) = p;
    p in {F: ex f st f is_a_proof_wrt_IPC X & Effect_IPC(f) = F} by A1;
    hence thesis by Th15;
end;
