reserve i for Integer,
  a, b, r, s for Real;

theorem
  for f being PartFunc of REAL,REAL st f = REAL --> r holds f|REAL is
  continuous
proof
  let f be PartFunc of REAL,REAL such that
A1: f = REAL --> r;
  f|REAL is constant
  proof
    reconsider r as Element of REAL by XREAL_0:def 1;
    take r;
    let c be Element of REAL;
    assume c in dom(f|REAL);
    thus (f|REAL).c = f.c
      .= r by A1,FUNCOP_1:7;
  end;
  hence thesis;
end;
