 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 Th10:
  f is_a_proof_wrt_IPC X & X c= Y implies f is_a_proof_wrt_IPC Y
proof
  assume that
A1: f is_a_proof_wrt_IPC X and
A2: X c= Y;
  thus f <> {} by A1;
  let n;
  assume
A3: 1 <= n & n <= len f; then
   A4: f,n is_a_correct_step_wrt_IPC X by A1;
  (f.n)`2 = 0 or ... or (f.n)`2 = 10 by A3,Th3;
  then per cases;
  case (f.n)`2 = 0; then
    (f.n)`1 in X by A4,Def3;
    hence thesis by A2;
  end;
  case (f.n)`2 = 1;
    hence thesis by A4,Def3;
  end;
  case (f.n)`2 = 2;
    hence thesis by A4,Def3;
  end;
  case (f.n)`2 = 3;
    hence thesis by A4,Def3;
  end;
  case (f.n)`2 = 4;
    hence thesis by A4,Def3;
  end;
  case (f.n)`2 = 5;
    hence thesis by A4,Def3;
  end;
  case (f.n)`2 = 6;
    hence thesis by A4,Def3;
  end;
  case (f.n)`2 = 7;
    hence thesis by A4,Def3;
  end;
  case (f.n)`2 = 8;
    hence thesis by A4,Def3;
  end;
  case (f.n)`2 = 9;
    hence thesis by A4,Def3;
  end;
  case (f.n)`2 = 10;
    hence thesis by A4,Def3;
  end;
end;
