theorem Th52:
f in x & g in x implies f a.e.= g,M &
  Integral(M,(abs f) to_power k) = Integral(M,(abs g) to_power k)
proof
   assume f in x & g in x; then
   f a.e.= g,M & f in Lp_Functions(M,k) & g in Lp_Functions(M,k) by Th50;
   hence thesis by Th48;
end;
