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;
reserve z for Complex;

theorem
  (n+1)|^3 + (n+2)|^3 + (n+3)|^3 + (n+4)|^3 <> (n+5)|^3
  proof
    assume
A1: (n+1)|^3 + (n+2)|^3 + (n+3)|^3 + (n+4)|^3 = (n+5)|^3;
    (n+1)|^3 = (n+1)*(n+1)*(n+1) & (n+2)|^3 = (n+2)*(n+2)*(n+2) &
    (n+3)|^3 = (n+3)*(n+3)*(n+3) & (n+4)|^3 = (n+4)*(n+4)*(n+4) &
    (n+5)|^3 = (n+5)*(n+5)*(n+5) by POLYEQ_5:2;
    then
A2: 3*(n*n*n+5*n*n+5*n) = 25 by A1;
    not 3 divides 3*8+1 by NAT_4:9;
    hence thesis by A2;
  end;
