thus Sum (digits (247,10)) = 13 by Th134; :: according to NUMBER11:def 1 :: thesis: ( 13 divides 247 & ( for m being Nat st Sum (digits (m,10)) = 13 & 13 divides m holds
247 <= m ) )
247 = 19 * 13 ;
hence 13 divides 247 by INT_1:def 3; :: thesis: for m being Nat st Sum (digits (m,10)) = 13 & 13 divides m holds
247 <= m
let m be Nat; :: thesis: ( Sum (digits (m,10)) = 13 & 13 divides m implies 247 <= m )
assume A1: ( Sum (digits (m,10)) = 13 & 13 divides m ) ; :: thesis: 247 <= m
then consider j being Nat such that
A2: m = 13 * j by NAT_D:def 3;
assume m < 247 ; :: thesis: contradiction
then 13 * j < 13 * 19 by A2;
then j < 18 + 1 by XREAL_1:64;
then j <= 18 by NAT_1:9;
then not not j = 0 & ... & not j = 18 ;

per cases then ( j = 0 or j = 1 or j = 2 or j = 3 or j = 4 or j = 5 or j = 6 or j = 7 or j = 8 or j = 9 or j = 10 or j = 11 or j = 12 or j = 13 or j = 14 or j = 15 or j = 16 or j = 17 or j = 18 ) ;

suppose j = 0 ; :: thesis: contradiction

then Sum (digits (m,10)) = 0 by A2, Th6;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 1 ; :: thesis: contradiction

then Sum (digits (m,10)) = 4 by A2, Th102;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 2 ; :: thesis: contradiction

then Sum (digits (m,10)) = 8 by A2, Th104;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 3 ; :: thesis: contradiction

then Sum (digits (m,10)) = 12 by A2, Th106;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 4 ; :: thesis: contradiction

then Sum (digits (m,10)) = 7 by A2, Th108;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 5 ; :: thesis: contradiction

then Sum (digits (m,10)) = 11 by A2, Th110;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 6 ; :: thesis: contradiction

then Sum (digits (m,10)) = 15 by A2, Th112;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 7 ; :: thesis: contradiction

then Sum (digits (m,10)) = 10 by A2, Th114;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 8 ; :: thesis: contradiction

then Sum (digits (m,10)) = 5 by A2, Th116;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 9 ; :: thesis: contradiction

then Sum (digits (m,10)) = 9 by A2, Th118;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 10 ; :: thesis: contradiction

then Sum (digits (m,10)) = 4 by A2, Th41;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 11 ; :: thesis: contradiction

then Sum (digits (m,10)) = 8 by A2, Th78;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 12 ; :: thesis: contradiction

then Sum (digits (m,10)) = 12 by A2, Th120;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 13 ; :: thesis: contradiction

then Sum (digits (m,10)) = 16 by A2, Th122;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 14 ; :: thesis: contradiction

then Sum (digits (m,10)) = 11 by A2, Th124;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 15 ; :: thesis: contradiction

then Sum (digits (m,10)) = 15 by A2, Th126;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 16 ; :: thesis: contradiction

then Sum (digits (m,10)) = 10 by A2, Th128;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 17 ; :: thesis: contradiction

then Sum (digits (m,10)) = 5 by A2, Th130;
hence contradiction by A1; :: thesis: verum

end;

suppose j = 18 ; :: thesis: contradiction

then Sum (digits (m,10)) = 9 by A2, Th132;
hence contradiction by A1; :: thesis: verum

end;

end;