
theorem
Sierp36 2398,22
proof
  thus Sum digits(2398,10) = 22 by Th919;
  2398=109*22;
  hence 22 divides 2398 by INT_1:def 3;
  let m be Nat;
  assume A1: Sum digits(m,10) = 22 & 22 divides m;
  then consider j being Nat such that
  A2: m=22*j by NAT_D:def 3;
  assume m < 2398;
  then 22*j < 22*109 by A2;
  then j < 108+1 by XREAL_1:64;
  then j <= 108 by NAT_1:9;
  then j <= 99 or (j >= 99+1 & j <= 108) by NAT_1:9;
  then (j=0 or ... or j=99) or (j=100 or ... or j=108);
  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) = 4 by A2,Th58;
    hence contradiction by A1;
  end;
  suppose j=2;
    then Sum digits(m,10) = 8 by A2,Th62;
    hence contradiction by A1;
  end;
  suppose j=3;
    then Sum digits(m,10) = 12 by A2,Th66;
    hence contradiction by A1;
  end;
  suppose j=4;
    then Sum digits(m,10) = 16 by A2,Th70;
    hence contradiction by A1;
  end;
  suppose j=5;
    then Sum digits(m,10) = 2 by A2,Th37;
    hence contradiction by A1;
  end;
  suppose j=6;
    then Sum digits(m,10) = 6 by A2,Th76;
    hence contradiction by A1;
  end;
  suppose j=7;
    then Sum digits(m,10) = 10 by A2,Th80;
    hence contradiction by A1;
  end;
  suppose j=8;
    then Sum digits(m,10) = 14 by A2,Th84;
    hence contradiction by A1;
  end;
  suppose j=9;
    then Sum digits(m,10) = 18 by A2,Th88;
    hence contradiction by A1;
  end;
  suppose j=10;
    then Sum digits(m,10) = 4 by A2,Th363;
    hence contradiction by A1;
  end;
  suppose j=11;
    then Sum digits(m,10) = 8 by A2,Th749;
    hence contradiction by A1;
  end;
  suppose j=12;
    then Sum digits(m,10) = 12 by A2,Th751;
    hence contradiction by A1;
  end;
  suppose j=13;
    then Sum digits(m,10) = 16 by A2,Th753;
    hence contradiction by A1;
  end;
  suppose j=14;
    then Sum digits(m,10) = 11 by A2,Th755;
    hence contradiction by A1;
  end;
  suppose j=15;
    then Sum digits(m,10) = 6 by A2,Th757;
    hence contradiction by A1;
  end;
  suppose j=16;
    then Sum digits(m,10) = 10 by A2,Th207;
    hence contradiction by A1;
  end;
  suppose j=17;
    then Sum digits(m,10) = 14 by A2,Th254;
    hence contradiction by A1;
  end;
  suppose j=18;
    then Sum digits(m,10) = 18 by A2,Th759;
    hence contradiction by A1;
  end;
  suppose j=19;
    then Sum digits(m,10) = 13 by A2,Th310;
    hence contradiction by A1;
  end;
  suppose j=20;
    then Sum digits(m,10) = 8 by A2,Th375;
    hence contradiction by A1;
  end;
  suppose j=21;
    then Sum digits(m,10) = 12 by A2,Th761;
    hence contradiction by A1;
  end;
  suppose j=22;
    then Sum digits(m,10) = 16 by A2,Th763;
    hence contradiction by A1;
  end;
  suppose j=23;
    then Sum digits(m,10) = 11 by A2,Th765;
    hence contradiction by A1;
  end;
  suppose j=24;
    then Sum digits(m,10) = 15 by A2,Th767;
    hence contradiction by A1;
  end;
  suppose j=25;
    then Sum digits(m,10) = 10 by A2,Th769;
    hence contradiction by A1;
  end;
  suppose j=26;
    then Sum digits(m,10) = 14 by A2,Th771;
    hence contradiction by A1;
  end;
  suppose j=27;
    then Sum digits(m,10) = 18 by A2,Th773;
    hence contradiction by A1;
  end;
  suppose j=28;
    then Sum digits(m,10) = 13 by A2,Th775;
    hence contradiction by A1;
  end;
  suppose j=29;
    then Sum digits(m,10) = 17 by A2,Th777;
    hence contradiction by A1;
  end;
  suppose j=30;
    then Sum digits(m,10) = 12 by A2,Th397;
    hence contradiction by A1;
  end;
  suppose j=31;
    then Sum digits(m,10) = 16 by A2,Th779;
    hence contradiction by A1;
  end;
  suppose j=32;
    then Sum digits(m,10) = 11 by A2,Th781;
    hence contradiction by A1;
  end;
  suppose j=33;
    then Sum digits(m,10) = 15 by A2,Th783;
    hence contradiction by A1;
  end;
  suppose j=34;
    then Sum digits(m,10) = 19 by A2,Th785;
    hence contradiction by A1;
  end;
  suppose j=35;
    then Sum digits(m,10) = 14 by A2,Th787;
    hence contradiction by A1;
  end;
  suppose j=36;
    then Sum digits(m,10) = 18 by A2,Th789;
    hence contradiction by A1;
  end;
  suppose j=37;
    then Sum digits(m,10) = 13 by A2,Th791;
    hence contradiction by A1;
  end;
  suppose j=38;
    then Sum digits(m,10) = 17 by A2,Th354;
    hence contradiction by A1;
  end;
  suppose j=39;
    then Sum digits(m,10) = 21 by A2,Th793;
    hence contradiction by A1;
  end;
  suppose j=40;
    then Sum digits(m,10) = 16 by A2,Th417;
    hence contradiction by A1;
  end;
  suppose j=41;
    then Sum digits(m,10) = 11 by A2,Th795;
    hence contradiction by A1;
  end;
  suppose j=42;
    then Sum digits(m,10) = 15 by A2,Th797;
    hence contradiction by A1;
  end;
  suppose j=43;
    then Sum digits(m,10) = 19 by A2,Th799;
    hence contradiction by A1;
  end;
  suppose j=44;
    then Sum digits(m,10) = 23 by A2,Th801;
    hence contradiction by A1;
  end;
  suppose j=45;
    then Sum digits(m,10) = 18 by A2,Th803;
    hence contradiction by A1;
  end;
  suppose j=46;
    then Sum digits(m,10) = 4 by A2,Th805;
    hence contradiction by A1;
  end;
  suppose j=47;
    then Sum digits(m,10) = 8 by A2,Th807;
    hence contradiction by A1;
  end;
  suppose j=48;
    then Sum digits(m,10) = 12 by A2,Th809;
    hence contradiction by A1;
  end;
  suppose j=49;
    then Sum digits(m,10) = 16 by A2,Th811;
    hence contradiction by A1;
  end;
  suppose j=50;
    then Sum digits(m,10) = 2 by A2,Th439;
    hence contradiction by A1;
  end;
  suppose j=51;
    then Sum digits(m,10) = 6 by A2,Th813;
    hence contradiction by A1;
  end;
  suppose j=52;
    then Sum digits(m,10) = 10 by A2,Th815;
    hence contradiction by A1;
  end;
  suppose j=53;
    then Sum digits(m,10) = 14 by A2,Th817;
    hence contradiction by A1;
  end;
  suppose j=54;
    then Sum digits(m,10) = 18 by A2,Th819;
    hence contradiction by A1;
  end;
  suppose j=55;
    then Sum digits(m,10) = 4 by A2,Th821;
    hence contradiction by A1;
  end;
  suppose j=56;
    then Sum digits(m,10) = 8 by A2,Th823;
    hence contradiction by A1;
  end;
  suppose j=57;
    then Sum digits(m,10) = 12 by A2,Th825;
    hence contradiction by A1;
  end;
  suppose j=58;
    then Sum digits(m,10) = 16 by A2,Th827;
    hence contradiction by A1;
  end;
  suppose j=59;
    then Sum digits(m,10) = 20 by A2,Th829;
    hence contradiction by A1;
  end;
  suppose j=60;
    then Sum digits(m,10) = 6 by A2,Th461;
    hence contradiction by A1;
  end;
  suppose j=61;
    then Sum digits(m,10) = 10 by A2,Th831;
    hence contradiction by A1;
  end;
  suppose j=62;
    then Sum digits(m,10) = 14 by A2,Th833;
    hence contradiction by A1;
  end;
  suppose j=63;
    then Sum digits(m,10) = 18 by A2,Th835;
    hence contradiction by A1;
  end;
  suppose j=64;
    then Sum digits(m,10) = 13 by A2,Th837;
    hence contradiction by A1;
  end;
  suppose j=65;
    then Sum digits(m,10) = 8 by A2,Th839;
    hence contradiction by A1;
  end;
  suppose j=66;
    then Sum digits(m,10) = 12 by A2,Th841;
    hence contradiction by A1;
  end;
  suppose j=67;
    then Sum digits(m,10) = 16 by A2,Th843;
    hence contradiction by A1;
  end;
  suppose j=68;
    then Sum digits(m,10) = 20 by A2,Th845;
    hence contradiction by A1;
  end;
  suppose j=69;
    then Sum digits(m,10) = 15 by A2,Th847;
    hence contradiction by A1;
  end;
  suppose j=70;
    then Sum digits(m,10) = 10 by A2,Th483;
    hence contradiction by A1;
  end;
  suppose j=71;
    then Sum digits(m,10) = 14 by A2,Th849;
    hence contradiction by A1;
  end;
  suppose j=72;
    then Sum digits(m,10) = 18 by A2,Th851;
    hence contradiction by A1;
  end;
  suppose j=73;
    then Sum digits(m,10) = 13 by A2,Th853;
    hence contradiction by A1;
  end;
  suppose j=74;
    then Sum digits(m,10) = 17 by A2,Th855;
    hence contradiction by A1;
  end;
  suppose j=75;
    then Sum digits(m,10) = 12 by A2,Th857;
    hence contradiction by A1;
  end;
  suppose j=76;
    then Sum digits(m,10) = 16 by A2,Th859;
    hence contradiction by A1;
  end;
  suppose j=77;
    then Sum digits(m,10) = 20 by A2,Th861;
    hence contradiction by A1;
  end;
  suppose j=78;
    then Sum digits(m,10) = 15 by A2,Th863;
    hence contradiction by A1;
  end;
  suppose j=79;
    then Sum digits(m,10) = 19 by A2,Th865;
    hence contradiction by A1;
  end;
  suppose j=80;
    then Sum digits(m,10) = 14 by A2,Th505;
    hence contradiction by A1;
  end;
  suppose j=81;
    then Sum digits(m,10) = 18 by A2,Th867;
    hence contradiction by A1;
  end;
  suppose j=82;
    then Sum digits(m,10) = 13 by A2,Th869;
    hence contradiction by A1;
  end;
  suppose j=83;
    then Sum digits(m,10) = 17 by A2,Th871;
    hence contradiction by A1;
  end;
  suppose j=84;
    then Sum digits(m,10) = 21 by A2,Th873;
    hence contradiction by A1;
  end;
  suppose j=85;
    then Sum digits(m,10) = 16 by A2,Th875;
    hence contradiction by A1;
  end;
  suppose j=86;
    then Sum digits(m,10) = 20 by A2,Th877;
    hence contradiction by A1;
  end;
  suppose j=87;
    then Sum digits(m,10) = 15 by A2,Th879;
    hence contradiction by A1;
  end;
  suppose j=88;
    then Sum digits(m,10) = 19 by A2,Th881;
    hence contradiction by A1;
  end;
  suppose j=89;
    then Sum digits(m,10) = 23 by A2,Th883;
    hence contradiction by A1;
  end;
  suppose j=90;
    then Sum digits(m,10) = 18 by A2,Th527;
    hence contradiction by A1;
  end;
  suppose j=91;
    then Sum digits(m,10) = 4 by A2,Th885;
    hence contradiction by A1;
  end;
  suppose j=92;
    then Sum digits(m,10) = 8 by A2,Th887;
    hence contradiction by A1;
  end;
  suppose j=93;
    then Sum digits(m,10) = 12 by A2,Th889;
    hence contradiction by A1;
  end;
  suppose j=94;
    then Sum digits(m,10) = 16 by A2,Th891;
    hence contradiction by A1;
  end;
  suppose j=95;
    then Sum digits(m,10) = 11 by A2,Th893;
    hence contradiction by A1;
  end;
  suppose j=96;
    then Sum digits(m,10) = 6 by A2,Th895;
    hence contradiction by A1;
  end;
  suppose j=97;
    then Sum digits(m,10) = 10 by A2,Th897;
    hence contradiction by A1;
  end;
  suppose j=98;
    then Sum digits(m,10) = 14 by A2,Th899;
    hence contradiction by A1;
  end;
  suppose j=99;
    then Sum digits(m,10) = 18 by A2,Th901;
    hence contradiction by A1;
  end;
  suppose j=100;
    then Sum digits(m,10) = 4 by A2,Th549;
    hence contradiction by A1;
  end;
  suppose j=101;
    then Sum digits(m,10) = 8 by A2,Th903;
    hence contradiction by A1;
  end;
  suppose j=102;
    then Sum digits(m,10) = 12 by A2,Th905;
    hence contradiction by A1;
  end;
  suppose j=103;
    then Sum digits(m,10) = 16 by A2,Th907;
    hence contradiction by A1;
  end;
  suppose j=104;
    then Sum digits(m,10) = 20 by A2,Th909;
    hence contradiction by A1;
  end;
  suppose j=105;
    then Sum digits(m,10) = 6 by A2,Th911;
    hence contradiction by A1;
  end;
  suppose j=106;
    then Sum digits(m,10) = 10 by A2,Th913;
    hence contradiction by A1;
  end;
  suppose j=107;
    then Sum digits(m,10) = 14 by A2,Th915;
    hence contradiction by A1;
  end;
  suppose j=108;
    then Sum digits(m,10) = 18 by A2,Th917;
    hence contradiction by A1;
  end;
end;
