reserve x,y for Real,
  i, j for non zero Element of NAT,
  I, O for non empty set,
  s,s1,s2,s3 for Element of I,
  w, w1, w2 for FinSequence of I,
  t for Element of O,
  S for non empty FSM over I,
  q, q1 for State of S;
reserve n, m, o, p for non zero Element of NAT,
  M for non empty Moore-SM_Final over I, O,
  q for State of M;

theorem
  for M being calculating_type halting
  non empty Moore-SM_Final over [:REAL, REAL:], succ REAL
  st the carrier of M = succ REAL & the FinalS of M = REAL &
  the InitS of M = REAL & the OFun of M = id the carrier of M &
  (for x,y st x < y holds (the Tran of M).[the InitS of M, [x,y]] = x) &
  (for x,y st x >= y holds (the Tran of M).[the InitS of M, [x,y]] = y)
  for x,y being Element of REAL holds Result([x,y], M) = min(x,y)
proof
  let M being calculating_type halting
  non empty Moore-SM_Final over [:REAL, REAL:], succ REAL;
  assume that
A1: the carrier of M = succ REAL and
A2: the FinalS of M = REAL and
A3: the InitS of M = REAL and
A4: the OFun of M = id the carrier of M and
A5: for x,y st x < y holds (the Tran of M).[the InitS of M, [x,y]] = x and
A6: for x,y st x >= y holds (the Tran of M).[the InitS of M, [x,y]] = y;
  let x,y be Element of REAL;
   min(x,y) in REAL by XREAL_0:def 1;
   then
A7: min(x,y) in succ REAL by XBOOLE_0:def 3;
  min(x,y) is_result_of [x,y], M by A1,A2,A3,A4,A5,A6,Th24;
  hence thesis by A7,Def9;
end;
