reserve i,n for Nat;
reserve m for non zero Nat;
reserve p,q for Tuple of n, BOOLEAN;
reserve d,d1,d2 for Element of BOOLEAN;

theorem Th7:
  Bin1 (m+1) = Bin1 (m)^<*FALSE*>
proof
  A1: len(Bin1(m+1)) = m+1 & len(Bin1(m)^<*FALSE*>) = m+1 by CARD_1:def 7;
 for i st i in Seg (m+1) holds (Bin1(m+1))/.i = (Bin1(m)^<*FALSE*>)/.i
  proof
    let i such that
A2: i in Seg (m+1);
    per cases by A2,FINSEQ_2:7;
    suppose
A3:   i in Seg m;
      thus (Bin1(m+1))/.i = (Bin1(m)^<*FALSE*>)/.i
      proof
        per cases;
        suppose
A4:       i = 1;
       (Bin1(m)^<*FALSE*>)/.i = (Bin1(m))/.i by A3,BINARITH:1
            .= TRUE by A3,A4,Th5;
          hence thesis by A2,A4,Th5;
        end;
        suppose
A5:       i <> 1;
       (Bin1(m)^<*FALSE*>)/.i = (Bin1(m))/.i by A3,BINARITH:1
            .= FALSE by A3,A5,Th6;
          hence thesis by A2,A5,Th6;
        end;
      end;
    end;
    suppose
A6:   i = m+1;
  1 <> m + 1 by NAT_1:14;
then   (Bin1(m+1))/.i = FALSE by A2,A6,Th6;
      hence thesis by A6,BINARITH:2;
    end;
  end;
  hence thesis by A1,Lm1;
end;
