reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem
  r|^3 + (r+1)|^3 + (r+2)|^3 = (r+3)|^3 iff r = 3
  proof
A1: r|^3 = r*r*r & (r+1)|^3 = (r+1)*(r+1)*(r+1) &
    (r+2)|^3 = (r+2)*(r+2)*(r+2) &
    (r+3)|^3 = (r+3)*(r+3)*(r+3) by POLYEQ_5:2;
    thus r|^3 + (r+1)|^3 + (r+2)|^3 = (r+3)|^3 implies r = 3
    proof
      set t = r-3;
      assume r|^3 + (r+1)|^3 + (r+2)|^3 = (r+3)|^3;
      then
A2:   (2*t)*(t*t+9*t+21) = 0 by A1;
      delta(1,9,21) < 0;
      then 1*t^2+9*t+21 > 0 by QUIN_1:3;
      then 2*t = 0 by A2;
      hence thesis;
    end;
    thus thesis by A1;
  end;
