assume 10 satisfies_Sierpinski_problem_86 ; :: thesis: contradiction

per cases then ( ( 10 - 1 is prime & ex x, y being Prime st
( x <> y & 10 + 1 = x * y ) ) or ( 10 + 1 is prime & ex x, y being Prime st
( x <> y & 10 - 1 = x * y ) ) ) by Th52;

suppose that 10 - 1 is prime and
A1: ex x, y being Prime st
( x <> y & 10 + 1 = x * y ) ; :: thesis: contradiction

thus contradiction by A1, XPRIMES1:11; :: thesis: verum

end;

suppose that 10 + 1 is prime and
A2: ex x, y being Prime st
( x <> y & 10 - 1 = x * y ) ; :: thesis: contradiction

consider x, y being Prime such that
A3: x <> y and
A4: x * y = 9 by A2;
x divides x * y ;
then ( x = 1 or x = 3 or x = 9 ) by A4, Th9;
hence contradiction by A3, A4; :: thesis: verum

end;

end;