 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 Th13:
  X c= {F: ex f st f is_a_proof_wrt_IPC X & Effect_IPC(f) = F}
proof
  let a be object;
  assume
A1: a in X;
  then reconsider p=a as Element of MC-wff;
  reconsider pp=[p,0] as Element of [:MC-wff,Proof_Step_Kinds_IPC:]
    by Th1,ZFMISC_1:87;
  set f=<*pp*>;
A2: len f = 1 by FINSEQ_1:40;
  (f.len f)`1 = p by A2; then
A4: Effect_IPC(f) = p by Def5;
  1 <= n & n <= len f implies f,n is_a_correct_step_wrt_IPC X
  proof
    assume 1 <= n & n <= len f; then
A5: n=1 by A2,XXREAL_0:1;
A6: (f.1)`2 = 0;
    (f.n)`1 in X by A1,A5;
    hence thesis by A5,A6,Def3;
  end; then
  f is_a_proof_wrt_IPC X;
  hence thesis by A4;
end;
