reserve a,b,c,d,x,j,k,l,m,n for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem Th10:
  (a-b)*(a|^m-b|^m) >= 0
  proof
    A0: a|^0 = 1 & b|^0 = 1 by NEWTON:4;
    per cases;
    suppose
      A1: a|^m - b|^m >= 0;
      then a|^m - b|^m + b|^m >= 0 + b|^m by XREAL_1:7;
      then a >= b or m < 1 by PREPOWER:10;
      then a-b >= b - b or m = 0 by XREAL_1:9,NAT_1:14;
      hence thesis by A0,A1;
    end;
    suppose
      A2: a|^m - b|^m < 0;
      then a|^m - b|^m + b|^m < 0 + b|^m by XREAL_1:8;
      then a - b < b - b by Lm3a,XREAL_1:9;
      hence thesis by A2;
    end;
  end;
