
theorem Th78:
  for f being Function, x, y being object st x in dom f & y in f.x
  holds renameElementsDistinctlyFunc(f,x).y = [f,x,y]
proof
  let f be Function, x, y be object;
  assume A1: x in dom f & y in f.x;
  then y in dom renameElementsDistinctlyFunc(f,x) by PARTFUN1:def 2;
  hence renameElementsDistinctlyFunc(f,x).y
     = [ (f.x --> [f,x]).y, id(f.x).y ] by FUNCT_3:def 7
    .= [ [f,x], id(f.x).y ] by A1, FUNCOP_1:7
    .= [ [f,x], y ] by A1, FUNCT_1:18
    .= [f,x,y] by XTUPLE_0:def 4;
end;
