 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 Th9:
  f is_a_proof_wrt_IPC X & g is_a_proof_wrt_IPC X implies f^g
  is_a_proof_wrt_IPC X
proof
  assume that
A1: f is_a_proof_wrt_IPC X and
A2: g is_a_proof_wrt_IPC X;
  thus f^g <> {} by A1;
  let n such that
A3: 1 <= n and
A4: n <= len(f^g);
 now per cases;
    suppose
A5:   n <= len f; then
      f,n is_a_correct_step_wrt_IPC X by A1,A3;
      hence thesis by A3,A5,Th7;
    end;
    suppose
A6:   len f < n;
      then reconsider k=n - len f as Element of NAT by NAT_1:21;
A7:  k + len f <= len g + len f by A4,FINSEQ_1:22;
  len f + 1 <= k + len f by A6,NAT_1:13;
then A8:  1 <= k by XREAL_1:6;
A9:  k <= len g by A7,XREAL_1:6; then
      k + len f = n & g,k is_a_correct_step_wrt_IPC X by A2,A8;
      hence thesis by A8,A9,Th8;
    end;
  end;
  hence thesis;
end;
