reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem
  not ex x,y being positive Nat st x*(x+1) = 4*y*(y+1)
  proof
    given x,y being positive Nat such that
A1: x*(x+1) = 4*y*(y+1);
    (2*(2*y+1))^2 - (2*x+1)^2 = (4*y-2*x+1)*(4*y+2*x+3);
    then
A2: (4*y+2*x+3) divides 3 by A1;
    0+3 < 4*y+2*x+3 by XREAL_1:8;
    hence contradiction by A2,NAT_D:7;
  end;
