reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem Th31:
  for a,b being non negative Real holds (a+b)|^(n+2)>=a|^
  (n+2)+(n+2)*(a|^(n+1))*b
proof
  let a,b be non negative Real;
  defpred X[Nat] means (a+b)|^($1+2)>=a|^($1+2)+($1+2)*(a|^($1+1))*
  b;
A1: for n st X[n] holds X[n+1]
  proof
    let n;
    a|^(n+1)>=0
    proof
      per cases;
      suppose
        a=0;
        hence thesis;
      end;
      suppose
        a>0;
        hence thesis by PREPOWER:6;
      end;
    end;
    then
A2: a|^(n+3)+(n+3)*(a|^(n+2))*b<=(n+2)*(a|^(n+1))*(b^2)+ (a|^(n+3)+(n+3)*
    (a|^(n+2))*b) by XREAL_1:31;
    assume (a+b)|^(n+2)>=a|^(n+2)+(n+2)*(a|^(n+1))*b;
    then (a+b)|^(n+2)*(a+b)>=(a|^(n+2)+(n+2)*(a|^(n+1))*b)*(a+b) by XREAL_1:64;
    then (a+b)|^(n+2+1)>=(a|^(n+2)+(n+2)*(a|^(n+1))*b)*(a+b) by NEWTON:6;
    then (a+b)|^(n+3)>=a|^(n+2)*a+b*a|^(n+2)+(n+2)*(a+b)*(a|^(n+1))*b;
    then (a+b)|^(n+3)>=a|^(n+2+1)+b*a|^(n+2)+(n+2)*(a+b)*(a|^(n+1))*b by
NEWTON:6;
    then
    (a+b)|^(n+3)>=a|^(n+3)+a|^(n+2)*b+(n+2)*(a*(a|^(n+1)))*b+ (n+2)*(a|^(
    n+1))*(b*b);
    then
    (a+b)|^(n+3)>=a|^(n+3)+a|^(n+2)*b+(n+2)*((a|^(n+1+1)))*b+ (n+2)*(a|^(
    n+1))*(b*b) by NEWTON:6;
    hence thesis by A2,XXREAL_0:2;
  end;
A3: a|^(0+2)+(0+2)*a|^(0+1)*b = a^2+2*a|^1*b by Lm1
    .=a^2+2*a*b;
  (a+b)|^(0+2) = (a+b)^2 by Lm1
    .=a^2+2*a*b+b^2;
  then
A4: X[0] by A3,XREAL_1:31;
  for n holds X[n] from NAT_1:sch 2(A4,A1);
  hence thesis;
end;
