reserve M for non empty MetrSpace,
  c,g1,g2 for Element of M;
reserve N for non empty MetrStruct,
  w for Element of N,
  G for Subset-Family of N,
  C for Subset of N;
reserve R for Reflexive non empty MetrStruct;
reserve T for Reflexive symmetric triangle non empty MetrStruct,
  t1 for Element of T,
  Y for Subset-Family of T,
  P for Subset of T;
reserve f for Function,
  n,m,p,n1,n2,k for Nat,
  r,s,L for Real,
  x,y for set;

theorem Th1:
  for L st 0<L & L<1 holds for n,m st n<=m holds L to_power m <= L to_power n
proof
  let L such that
A1: 0<L & L<1;
  let n,m such that
A2: n<=m;
  per cases by A2,XXREAL_0:1;
  suppose n<m;
    hence thesis by A1,POWER:40;
  end;
  suppose n=m;
    hence thesis;
  end;
end;
