 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 Th3:
  for n being Nat holds 1 <= n & n <= len f implies
  (f.n)`2 = 0 or ... or (f.n)`2 = 10
proof
  let n be Nat;
  assume 1 <= n & n <= len f;
  then n in dom f by FINSEQ_3:25;
  then
  rng f c= [:MC-wff,Proof_Step_Kinds_IPC:] & f.n in rng f
    by FINSEQ_1:def 4,FUNCT_1:def 3;
  then (f.n)`2 in Proof_Step_Kinds_IPC by MCART_1:10;
  then ex k st k = (f.n)`2 & k <= 10;
  hence thesis;
end;
