theorem Th32:
  X common_on_dom H implies for x st x in X holds (r(#)H)#x = r(#) (H#x)
proof
  assume
A1: X common_on_dom H;
  let x;
  assume x in X;
  then {x} common_on_dom H by A1,Th25;
  hence thesis by Th29;
end;
