
theorem Th22:
  for n,m be Nat st n <= m for x,y being FinSequence holds
  InnerVertices (n-BitGFA1Str(x,y)) c= InnerVertices (m-BitGFA1Str(x,y))
proof
  let n,m be Nat such that
A1: n <= m;
  let x,y be FinSequence;
  consider i being Nat such that
A2: m = n+i by A1,NAT_1:10;
  defpred L[Nat] means InnerVertices (n-BitGFA1Str(x,y)) c=
  InnerVertices ((n+$1)-BitGFA1Str(x,y));
A3: L[ 0 ];
A4: for j being Nat st L[j] holds L[j+1]
  proof
    let j be Nat;
    set Sn = n-BitGFA1Str(x,y);
    set Snj = (n+j)-BitGFA1Str(x,y);
    set SSnj = BitGFA1Str(x .((n+j)+1), y.((n+j)+1),
    (n+j)-BitGFA1CarryOutput(x, y));
    assume
A5: InnerVertices (Sn) c= InnerVertices (Snj);
A6: InnerVertices (Sn) c= InnerVertices (Sn) \/ InnerVertices (SSnj)
    by XBOOLE_1:7;
    InnerVertices (Sn) \/ InnerVertices (SSnj) c=
    InnerVertices (Snj) \/ InnerVertices (SSnj) by A5,XBOOLE_1:9;
    then InnerVertices (Sn) c= InnerVertices (Snj) \/ InnerVertices (SSnj)
    by A6,XBOOLE_1:1;
    then InnerVertices (Sn) c= InnerVertices (Snj +* SSnj) by FACIRC_1:27;
    hence thesis by Th21;
  end;
  for j being Nat holds L[j] from NAT_1:sch 2(A3,A4);
  hence thesis by A2;
end;
