
theorem Th11:
  for f being Function, i, j being object st i in dom f & j in dom f
  holds f = f +* (i,j) --> (f.i,f.j)
proof
  let f be Function, i, j be object;
  assume A1: i in dom f & j in dom f;
  thus f = f +* (j .--> f.j) by A1, FUNCT_4:7, LATTICE2:5
    .= (f +* (i .--> f.i)) +* (j .--> f.j) by A1, FUNCT_4:7, LATTICE2:5
    .= f +* ((i .--> f.i) +* (j .--> f.j)) by FUNCT_4:14
    .= f +* (i,j) --> (f.i,f.j) by FUNCT_4:def 4;
end;
