given a, b, c being Prime such that A1: a,b,c are_mutually_distinct and
A2: (18 |^ 2) + 1 = (a * b) * c ; :: according to NUMBER07:def 2 :: thesis: contradiction
18 |^ 2 = 18 * 18 by WSIERP_1:1;

per cases then ( ( a * b = 1 & c = 325 ) or ( a * b = 5 & c = 65 ) or ( a * b = 13 & c = 25 ) or ( a * b = 25 & c = 13 ) or ( a * b = 65 & c = 5 ) or ( a * b = 325 & c = 1 ) ) by A2, Th36;

suppose ( a * b = 1 & c = 325 ) ; :: thesis: contradiction

hence contradiction by XPRIMES0:325; :: thesis: verum

end;

suppose ( a * b = 5 & c = 65 ) ; :: thesis: contradiction

hence contradiction by XPRIMES1:5; :: thesis: verum

end;

suppose ( a * b = 13 & c = 25 ) ; :: thesis: contradiction

hence contradiction by XPRIMES1:13; :: thesis: verum

end;

suppose ( a * b = 25 & c = 13 ) ; :: thesis: contradiction

then ( ( a = 1 & b = 25 ) or ( a = 5 & b = 5 ) or ( a = 25 & b = 1 ) ) by Th28;
hence contradiction by A1, XPRIMES0:1; :: thesis: verum

end;

suppose that A3: a * b = 65 and
A4: c = 5 ; :: thesis: contradiction

( ( a = 1 & b = 65 ) or ( a = 5 & b = 13 ) or ( a = 13 & b = 5 ) or ( a = 65 & b = 1 ) ) by A3, Th31;
hence contradiction by A1, A4, XPRIMES0:1; :: thesis: verum

end;

suppose ( a * b = 325 & c = 1 ) ; :: thesis: contradiction

hence contradiction by XPRIMES0:1; :: thesis: verum

end;

end;