theorem
  x in dom f1 & x in dom f2 implies
  for y1,y2 holds <:f1,f2:>.x = [y1,y2] iff <:<*f1,f2*>:>.x = <*y1,y2*>
proof
A1: <*f1.x,f2.x*>.1 = f1.x & <*f1.x,f2.x*>.2 = f2.x;
  assume x in dom f1 & x in dom f2;
  then
A2: x in dom f1 /\ dom f2 by XBOOLE_0:def 4;
  let y1,y2;
A3: <*y1,y2*>.1 = y1 & <*y1,y2*>.2 = y2;
  [f1.x,f2.x] = [y1,y2] iff f1.x = y1 & f2.x = y2 by XTUPLE_0:1;
  hence thesis by A2,A1,A3,Th140,FUNCT_3:48;
end;
