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 Th6:
  a|^m + b|^m <= c|^m implies a <= c
  proof
    assume
    A0: a|^m + b|^m <= c|^m;
    m=0 implies a|^m + b|^m > c|^m by Lm0; then
    A2e: 0+1 <= m by A0,NAT_1:13;
    a|^m + b|^m <= c|^m + b|^m by A0,NAT_1:12;
    then a|^m + b|^m -b|^m <= c|^m + b|^m- b|^m by XREAL_1:9;
    hence thesis by A2e,PREPOWER:10;
  end;
