let M be non empty calculating_type halting Moore-SM_Final over [:REAL,REAL:], succ REAL; :: thesis: ( 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 being Real st ( x > 0 or y > 0 ) holds
the Tran of M . [ the InitS of M,[x,y]] = 1 ) & ( for x, y being Real st ( x = 0 or y = 0 ) & x <= 0 & y <= 0 holds
the Tran of M . [ the InitS of M,[x,y]] = 0 ) & ( for x, y being Real st x < 0 & y < 0 holds
the Tran of M . [ the InitS of M,[x,y]] = - 1 ) implies for x, y being Element of REAL holds Result ([x,y],M) = max ((sgn x),(sgn y)) )
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 ; :: thesis: ( ex x, y being Real st
( ( x > 0 or y > 0 ) & not the Tran of M . [ the InitS of M,[x,y]] = 1 ) or ex x, y being Real st
( ( x = 0 or y = 0 ) & x <= 0 & y <= 0 & not the Tran of M . [ the InitS of M,[x,y]] = 0 ) or ex x, y being Real st
( x < 0 & y < 0 & not the Tran of M . [ the InitS of M,[x,y]] = - 1 ) or for x, y being Element of REAL holds Result ([x,y],M) = max ((sgn x),(sgn y)) )
assume that
A5: for x, y being Real st ( x > 0 or y > 0 ) holds
the Tran of M . [ the InitS of M,[x,y]] = 1 and
A6: for x, y being Real st ( x = 0 or y = 0 ) & x <= 0 & y <= 0 holds
the Tran of M . [ the InitS of M,[x,y]] = 0 and
A7: for x, y being Real st x < 0 & y < 0 holds
the Tran of M . [ the InitS of M,[x,y]] = - 1 ; :: thesis: for x, y being Element of REAL holds Result ([x,y],M) = max ((sgn x),(sgn y))
let x, y be Element of REAL ; :: thesis: Result ([x,y],M) = max ((sgn x),(sgn y))
max ((sgn x),(sgn y)) in REAL by XREAL_0:def 1;
then A8: max ((sgn x),(sgn y)) in succ REAL by XBOOLE_0:def 3;
max ((sgn x),(sgn y)) is_result_of [x,y],M by A1, A2, A3, A4, A5, A6, A7, Th26;
hence Result ([x,y],M) = max ((sgn x),(sgn y)) by A8, Def9; :: thesis: verum