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 Th23:
  for M being non empty Moore-SM_Final over [:REAL, REAL:], succ REAL
  st M is calculating_type & 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 max(x,y) is_result_of [x,y], M
proof
  deffunc F(Real,Real) = In(max($1,$2),REAL);
  consider f being BinOp of REAL such that
A1: for x,y being Element of REAL holds f.(x,y) = F(x,y)
   from BINOP_1:sch 4;
A2: for x,y being Element of REAL holds f.(x,y) = max(x,y)
   proof let x,y be Element of REAL;
     f.(x,y) = F(x,y) by A1;
    hence thesis;
   end;
  let M being non empty Moore-SM_Final over [:REAL, REAL:], succ REAL;
  assume that
A3: M is calculating_type and
A4: the carrier of M = succ REAL and
A5: the FinalS of M = REAL and
A6: the InitS of M = REAL and
A7: the OFun of M = id the carrier of M;
  assume that
A8: for x,y st x >= y holds (the Tran of M).[the InitS of M, [x,y]] = x and
A9: 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;
   reconsider x,y as Element of REAL;
  now
    let x,y be Element of REAL;
A10: x >= y implies (the Tran of M).[the InitS of M, [x,y]] = x by A8;
    x < y implies (the Tran of M).[the InitS of M, [x,y]] = y by A9;
    then (the Tran of M).[the InitS of M, [x,y]] = max(x,y) by A10,
XXREAL_0:def 10;
    hence (the Tran of M).[the InitS of M, [x,y]] = f.(x,y) by A2;
  end;
  then f.(x,y) is_result_of [x,y], M by A3,A4,A5,A6,A7,Th22;
  hence thesis by A2;
end;
