 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 Th15:
  for X holds {p: ex f st f is_a_proof_wrt_IPC X &
  Effect_IPC(f) = p} = CnIPC(X)
proof
  let X;
  set PX = {p: ex f st f is_a_proof_wrt_IPC X & Effect_IPC(f) = p};
A1: PX c= CnIPC(X) by Lm12;
  reconsider PX as Subset of MC-wff by Lm1;
  X c= PX by Th13;
  hence thesis by A1,Th14,INTPRO_1:11;
end;
