
theorem
Sierp36 1679,23
proof
  thus Sum digits(1679,10) = 23 by Th1046;
  1679=73*23;
  hence 23 divides 1679 by INT_1:def 3;
  let m be Nat;
  assume A1: Sum digits(m,10) = 23 & 23 divides m;
  then consider j being Nat such that
  A2: m=23*j by NAT_D:def 3;
  assume m < 1679;
  then 23*j < 23*73 by A2;
  then j < 72+1 by XREAL_1:64;
  then j <= 72 by NAT_1:9;
  then j=0 or ... or j=72;
  then per cases;
  suppose j=0;
    then Sum digits(m,10) = 0 by A2,Th6;
    hence contradiction by A1;
  end;
  suppose j=1;
    then Sum digits(m,10) = 5 by A2,Th922;
    hence contradiction by A1;
  end;
  suppose j=2;
    then Sum digits(m,10) = 10 by A2,Th924;
    hence contradiction by A1;
  end;
  suppose j=3;
    then Sum digits(m,10) = 15 by A2,Th926;
    hence contradiction by A1;
  end;
  suppose j=4;
    then Sum digits(m,10) = 11 by A2,Th928;
    hence contradiction by A1;
  end;
  suppose j=5;
    then Sum digits(m,10) = 7 by A2,Th930;
    hence contradiction by A1;
  end;
  suppose j=6;
    then Sum digits(m,10) = 12 by A2,Th932;
    hence contradiction by A1;
  end;
  suppose j=7;
    then Sum digits(m,10) = 8 by A2,Th934;
    hence contradiction by A1;
  end;
  suppose j=8;
    then Sum digits(m,10) = 13 by A2,Th936;
    hence contradiction by A1;
  end;
  suppose j=9;
    then Sum digits(m,10) = 9 by A2,Th938;
    hence contradiction by A1;
  end;
  suppose j=10;
    then Sum digits(m,10) = 5 by A2,Th940;
    hence contradiction by A1;
  end;
  suppose j=11;
    then Sum digits(m,10) = 10 by A2,Th942;
    hence contradiction by A1;
  end;
  suppose j=12;
    then Sum digits(m,10) = 15 by A2,Th944;
    hence contradiction by A1;
  end;
  suppose j=13;
    then Sum digits(m,10) = 20 by A2,Th946;
    hence contradiction by A1;
  end;
  suppose j=14;
    then Sum digits(m,10) = 7 by A2,Th948;
    hence contradiction by A1;
  end;
  suppose j=15;
    then Sum digits(m,10) = 12 by A2,Th950;
    hence contradiction by A1;
  end;
  suppose j=16;
    then Sum digits(m,10) = 17 by A2,Th209;
    hence contradiction by A1;
  end;
  suppose j=17;
    then Sum digits(m,10) = 13 by A2,Th256;
    hence contradiction by A1;
  end;
  suppose j=18;
    then Sum digits(m,10) = 9 by A2,Th952;
    hence contradiction by A1;
  end;
  suppose j=19;
    then Sum digits(m,10) = 14 by A2,Th312;
    hence contradiction by A1;
  end;
  suppose j=20;
    then Sum digits(m,10) = 10 by A2,Th377;
    hence contradiction by A1;
  end;
  suppose j=21;
    then Sum digits(m,10) = 15 by A2,Th954;
    hence contradiction by A1;
  end;
  suppose j=22;
    then Sum digits(m,10) = 11 by A2,Th765;
    hence contradiction by A1;
  end;
  suppose j=23;
    then Sum digits(m,10) = 16 by A2,Th956;
    hence contradiction by A1;
  end;
  suppose j=24;
    then Sum digits(m,10) = 12 by A2,Th958;
    hence contradiction by A1;
  end;
  suppose j=25;
    then Sum digits(m,10) = 17 by A2,Th960;
    hence contradiction by A1;
  end;
  suppose j=26;
    then Sum digits(m,10) = 22 by A2,Th962;
    hence contradiction by A1;
  end;
  suppose j=27;
    then Sum digits(m,10) = 9 by A2,Th964;
    hence contradiction by A1;
  end;
  suppose j=28;
    then Sum digits(m,10) = 14 by A2,Th966;
    hence contradiction by A1;
  end;
  suppose j=29;
    then Sum digits(m,10) = 19 by A2,Th968;
    hence contradiction by A1;
  end;
  suppose j=30;
    then Sum digits(m,10) = 15 by A2,Th970;
    hence contradiction by A1;
  end;
  suppose j=31;
    then Sum digits(m,10) = 11 by A2,Th972;
    hence contradiction by A1;
  end;
  suppose j=32;
    then Sum digits(m,10) = 16 by A2,Th974;
    hence contradiction by A1;
  end;
  suppose j=33;
    then Sum digits(m,10) = 21 by A2,Th976;
    hence contradiction by A1;
  end;
  suppose j=34;
    then Sum digits(m,10) = 17 by A2,Th978;
    hence contradiction by A1;
  end;
  suppose j=35;
    then Sum digits(m,10) = 13 by A2,Th980;
    hence contradiction by A1;
  end;
  suppose j=36;
    then Sum digits(m,10) = 18 by A2,Th982;
    hence contradiction by A1;
  end;
  suppose j=37;
    then Sum digits(m,10) = 14 by A2,Th984;
    hence contradiction by A1;
  end;
  suppose j=38;
    then Sum digits(m,10) = 19 by A2,Th358;
    hence contradiction by A1;
  end;
  suppose j=39;
    then Sum digits(m,10) = 24 by A2,Th986;
    hence contradiction by A1;
  end;
  suppose j=40;
    then Sum digits(m,10) = 11 by A2,Th421;
    hence contradiction by A1;
  end;
  suppose j=41;
    then Sum digits(m,10) = 16 by A2,Th988;
    hence contradiction by A1;
  end;
  suppose j=42;
    then Sum digits(m,10) = 21 by A2,Th990;
    hence contradiction by A1;
  end;
  suppose j=43;
    then Sum digits(m,10) = 26 by A2,Th992;
    hence contradiction by A1;
  end;
  suppose j=44;
    then Sum digits(m,10) = 4 by A2,Th805;
    hence contradiction by A1;
  end;
  suppose j=45;
    then Sum digits(m,10) = 9 by A2,Th994;
    hence contradiction by A1;
  end;
  suppose j=46;
    then Sum digits(m,10) = 14 by A2,Th996;
    hence contradiction by A1;
  end;
  suppose j=47;
    then Sum digits(m,10) = 10 by A2,Th998;
    hence contradiction by A1;
  end;
  suppose j=48;
    then Sum digits(m,10) = 6 by A2,Th1000;
    hence contradiction by A1;
  end;
  suppose j=49;
    then Sum digits(m,10) = 11 by A2,Th1002;
    hence contradiction by A1;
  end;
  suppose j=50;
    then Sum digits(m,10) = 7 by A2,Th1004;
    hence contradiction by A1;
  end;
  suppose j=51;
    then Sum digits(m,10) = 12 by A2,Th1006;
    hence contradiction by A1;
  end;
  suppose j=52;
    then Sum digits(m,10) = 17 by A2,Th1008;
    hence contradiction by A1;
  end;
  suppose j=53;
    then Sum digits(m,10) = 13 by A2,Th1010;
    hence contradiction by A1;
  end;
  suppose j=54;
    then Sum digits(m,10) = 9 by A2,Th1012;
    hence contradiction by A1;
  end;
  suppose j=55;
    then Sum digits(m,10) = 14 by A2,Th1014;
    hence contradiction by A1;
  end;
  suppose j=56;
    then Sum digits(m,10) = 19 by A2,Th1016;
    hence contradiction by A1;
  end;
  suppose j=57;
    then Sum digits(m,10) = 6 by A2,Th1018;
    hence contradiction by A1;
  end;
  suppose j=58;
    then Sum digits(m,10) = 11 by A2,Th1020;
    hence contradiction by A1;
  end;
  suppose j=59;
    then Sum digits(m,10) = 16 by A2,Th1022;
    hence contradiction by A1;
  end;
  suppose j=60;
    then Sum digits(m,10) = 12 by A2,Th467;
    hence contradiction by A1;
  end;
  suppose j=61;
    then Sum digits(m,10) = 8 by A2,Th1024;
    hence contradiction by A1;
  end;
  suppose j=62;
    then Sum digits(m,10) = 13 by A2,Th1026;
    hence contradiction by A1;
  end;
  suppose j=63;
    then Sum digits(m,10) = 18 by A2,Th1028;
    hence contradiction by A1;
  end;
  suppose j=64;
    then Sum digits(m,10) = 14 by A2,Th1030;
    hence contradiction by A1;
  end;
  suppose j=65;
    then Sum digits(m,10) = 19 by A2,Th1032;
    hence contradiction by A1;
  end;
  suppose j=66;
    then Sum digits(m,10) = 15 by A2,Th847;
    hence contradiction by A1;
  end;
  suppose j=67;
    then Sum digits(m,10) = 11 by A2,Th1034;
    hence contradiction by A1;
  end;
  suppose j=68;
    then Sum digits(m,10) = 16 by A2,Th1036;
    hence contradiction by A1;
  end;
  suppose j=69;
    then Sum digits(m,10) = 21 by A2,Th1038;
    hence contradiction by A1;
  end;
  suppose j=70;
    then Sum digits(m,10) = 8 by A2,Th1040;
    hence contradiction by A1;
  end;
  suppose j=71;
    then Sum digits(m,10) = 13 by A2,Th1042;
    hence contradiction by A1;
  end;
  suppose j=72;
    then Sum digits(m,10) = 18 by A2,Th1044;
    hence contradiction by A1;
  end;
end;
