
theorem :: Problem 56
  for i being Nat holds ArProg (2,3).i is non triangular
  proof
    let i be Nat;
    assume ArProg (2,3).i is triangular; then
    consider k being Nat such that
A1: ArProg (2,3).i = Triangle k by NUMPOLY1:def 2;
    per cases by NUMBER02:23;
    suppose ex u being Nat st k = 3 * u + 1; then
      consider u being Nat such that
E1:   k = 3 * u + 1;
      3 divides Triangle (3 * u + 1) - 1 by Not3DividesTriangle;
      hence thesis by LemmaBono,A1,E1;
    end;
    suppose ex u being Nat st k = 3 * u + 2; then
      consider u being Nat such that
D1:   k = 3 * u + 2;
      thus thesis by A1,Lemma3ArProg,D1,Divides3Triangle2;
    end;
    suppose ex u being Nat st k = 3 * u; then
      consider u being Nat such that
C1:   k = 3 * u;
      thus thesis by A1,Lemma3ArProg,C1,Divides3Triangle;
    end;
  end;
