 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
  f is_a_proof_wrt_IPC X implies (f.1)`2 = 0 or ... or (f.1)`2 = 10
proof
  assume
A1: f is_a_proof_wrt_IPC X;
  then A2: 1 <= len f by Th5;
  then A3: f,1 is_a_correct_step_wrt_IPC X by A1;
  assume (f.1)`2 <> 0 & ... & (f.1)`2 <> 10;
  then per cases by A2,Th3;
  suppose (f.1)`2 = 10;
  then ex i,j,p,q st 1 <= i & i < 1 & 1 <= j & j < i &
  p = (f.j)`1 & q = (f.1)`1 & (f.i)`1 = p => q by A3,Def3;
  hence contradiction;
  end;
end;
