reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r for Real;
reserve p,p1,p2 for Prime;

theorem
  not 17 satisfies_Sierpinski_problem_48
  proof
    given a,b,c being Nat such that
A1: 17 = a+b+c and
A2: a > 1 and
A3: b > 1 and
A4: c > 1 and
A5: a,b,c are_mutually_coprime;
A6: 17 = 2*8+1;
    then a is odd by A1,A5,Th7;
    then
A7: a = 3 or a = 5 or a = 7 or a = 9 or a = 11 or a = 13
    by A1,A2,A3,A4,Lm4,Lm3;
    b is odd by A1,A5,A6,Th7;
    then
A8: b = 3 or b = 5 or b = 7 or b = 9 or b = 11 or b = 13
    by A1,A2,A3,A4,Lm4,Lm3;
A9: c is odd by A1,A5,A6,Th7;
    a+b+c = c+a+b;
    then
A10: c = 3 or c = 5 or c = 7 or c = 9 or c = 11 or c = 13
    by A1,A2,A3,A4,A9,Lm4,Lm3;
    not 3,3*3 are_coprime by Th1;
    hence contradiction by A1,A5,A7,A8,A10,EULER_1:1;
  end;
