reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem
  for x being Real holds
  (x+1)|^3 + (x+2)|^3 + (x+3)|^3 + (x+4)|^3 = (x+10)|^3 iff x = 10
  proof
    let x be Real;
A1: (x+1)|^3 = (x+1)*(x+1)*(x+1) by POLYEQ_5:2;
A2: (x+2)|^3 = (x+2)*(x+2)*(x+2) by POLYEQ_5:2;
A3: (x+3)|^3 = (x+3)*(x+3)*(x+3) by POLYEQ_5:2;
A4: (x+4)|^3 = (x+4)*(x+4)*(x+4) by POLYEQ_5:2;
A5: (x+10)|^3 = (x+10)*(x+10)*(x+10) by POLYEQ_5:2;
    thus (x+1)|^3 + (x+2)|^3 + (x+3)|^3 + (x+4)|^3 = (x+10)|^3 implies x = 10
    proof
      set t = x-10;
      assume (x+1)|^3 + (x+2)|^3 + (x+3)|^3 + (x+4)|^3 = (x+10)|^3;
      then t*(t*t+30*t+230) = 0 by A1,A2,A3,A4,A5;
      then
A6:   t = 0 or 1*t^2+30*t+230 = 0;
      delta(1,30,230) = -20;
      hence thesis by A6,QUIN_1:3;
    end;
    thus thesis by A1,A2,A3,A4,A5;
  end;
