theorem
  x in dom(f /\ g) implies (f /\ g).x = f.x
proof
  set y = (f /\ g).x;
  assume x in dom(f /\ g);
  then [x,y] in (f /\ g) by FUNCT_1:def 2;
  then [x,y] in f by XBOOLE_0:def 4;
  hence thesis by FUNCT_1:1;
end;
