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
  (a|^n+b|^n)/2 >= ((a+b)/2)|^n
proof
  defpred X[Nat] means (a|^$1+b|^$1)/2 >= ((a+b)/2)|^$1;
A1: for n st X[n] holds X[n+1]
  proof
    let n;
    assume X[n];
    then
A2: ((a|^n+b|^n)/2)*(a+b)>= (((a+b)/2)|^n)*(a+b) by XREAL_1:64;
    per cases;
    suppose
A3:   a-b>=0;
      then a-b+b>=0+b by XREAL_1:6;
      then a|^n >= b|^n by PREPOWER:9;
      then a|^n-b|^n>=0 by XREAL_1:48;
      then (a-b)*(a|^n-b|^n)>=0 by A3;
      then a|^n*a-a|^n*b-(b|^n*a-b|^n*b)>=0;
      then a|^(n+1)-a|^n*b-(b|^n*a-b|^n*b)>=0 by NEWTON:6;
      then a|^(n+1)-a|^n*b-b|^n*a+b|^n*b>=0;
      then a|^(n+1)-a|^n*b-b|^n*a+b|^(n+1)>=0 by NEWTON:6;
      then a|^(n+1)-a|^n*b-b|^n*a+b|^(n+1)+(a|^n*b+b|^n*a)>=0+(a|^n*b+b|^n*a)
      by XREAL_1:6;
      then
      a|^(n+1)+b|^(n+1)+(a|^(n+1)+b|^(n+1))>=a|^n*b+b|^n*a+(a|^(n+1)+b|^(
      n+ 1)) by XREAL_1:6;
      then 2*a|^(n+1)+2*b|^(n+1)>=a|^(n+1)+a|^n*b+b|^n*a+b|^(n+1);
      then 2*a|^(n+1)+2*b|^(n+1)>=a|^n*a+a|^n*b+b|^n*a+b|^(n+1) by NEWTON:6;
      then 2*a|^(n+1)+2*b|^(n+1)>=a|^n*a+a|^n*b+b|^n*a+b|^n*b by NEWTON:6;
      then (2*(a|^(n+1)+b|^(n+1)))/2>=((a|^n+b|^n)*(a+b))/2 by XREAL_1:72;
      then (a|^(n+1)+b|^(n+1))>=(((a+b)/2)|^n)*(a+b) by A2,XXREAL_0:2;
      then (a|^(n+1)+b|^(n+1))>=((a+b)|^n/2|^n)*(a+b) by PREPOWER:8;
      then (a|^(n+1)+b|^(n+1))>=((a+b)|^n*(a+b))/2|^n;
      then (a|^(n+1)+b|^(n+1))>=((a+b)|^(n+1))/2|^n by NEWTON:6;
      then (a|^(n+1)+b|^(n+1)) /2>=(((a+b)|^(n+1))/2|^n)/2 by XREAL_1:72;
      then (a|^(n+1)+b|^(n+1)) /2>=((a+b)|^(n+1))/(2|^n*2) by XCMPLX_1:78;
      then (a|^(n+1)+b|^(n+1)) /2>=((a+b)|^(n+1))/(2|^(n+1)) by NEWTON:6;
      hence thesis by PREPOWER:8;
    end;
    suppose
A4:   a-b<0;
      then a-b+b<0+b by XREAL_1:6;
      then a|^n <= b|^n by PREPOWER:9;
      then a|^n-b|^n<=0 by XREAL_1:47;
      then (a-b)*(a|^n-b|^n)>=0 by A4;
      then a|^n*a-a|^n*b-(b|^n*a-b|^n*b)>=0;
      then a|^(n+1)-a|^n*b-(b|^n*a-b|^n*b)>=0 by NEWTON:6;
      then a|^(n+1)-a|^n*b-b|^n*a+b|^n*b>=0;
      then a|^(n+1)-a|^n*b-b|^n*a+b|^(n+1)>=0 by NEWTON:6;
      then a|^(n+1)-a|^n*b-b|^n*a+b|^(n+1)+(a|^n*b+b|^n*a)>=0+(a|^n*b+b|^n*a)
      by XREAL_1:6;
      then
      a|^(n+1)+b|^(n+1)+(a|^(n+1)+b|^(n+1))>=a|^n*b+b|^n*a+(a|^(n+1)+b|^(
      n+ 1)) by XREAL_1:6;
      then 2*a|^(n+1)+2*b|^(n+1)>=a|^(n+1)+a|^n*b+b|^n*a+b|^(n+1);
      then 2*a|^(n+1)+2*b|^(n+1)>=a|^n*a+a|^n*b+b|^n*a+b|^(n+1) by NEWTON:6;
      then 2*a|^(n+1)+2*b|^(n+1)>=a|^n*a+a|^n*b+b|^n*a+b|^n*b by NEWTON:6;
      then (2*(a|^(n+1)+b|^(n+1)))/2>=((a|^n+b|^n)*(a+b))/2 by XREAL_1:72;
      then (a|^(n+1)+b|^(n+1))>=(((a+b)/2)|^n)*(a+b) by A2,XXREAL_0:2;
      then (a|^(n+1)+b|^(n+1))>=((a+b)|^n/2|^n)*(a+b) by PREPOWER:8;
      then (a|^(n+1)+b|^(n+1))>=((a+b)|^n*(a+b))/2|^n;
      then (a|^(n+1)+b|^(n+1))>=((a+b)|^(n+1))/2|^n by NEWTON:6;
      then (a|^(n+1)+b|^(n+1)) /2>=(((a+b)|^(n+1))/2|^n)/2 by XREAL_1:72;
      then (a|^(n+1)+b|^(n+1)) /2>=((a+b)|^(n+1))/(2|^n*2) by XCMPLX_1:78;
      then (a|^(n+1)+b|^(n+1)) /2>=((a+b)|^(n+1))/(2|^(n+1)) by NEWTON:6;
      hence thesis by PREPOWER:8;
    end;
  end;
  (a|^0+b|^0)/2 = (1+b|^0)/2 by NEWTON:4
    .= (1+1)/2 by NEWTON:4
    .= ((a+b)/2)|^0 by NEWTON:4;
  then
A5: X[0];
  for n holds X[n] from NAT_1:sch 2(A5,A1);
  hence thesis;
end;
