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 < y holds

the Tran of M . [ the InitS of M,[x,y]] = x ) & ( for x, y being Real st x >= y holds

the Tran of M . [ the InitS of M,[x,y]] = y ) implies for x, y being Element of REAL holds Result ([x,y],M) = min (x,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 and

A5: for x, y being Real st x < y holds

the Tran of M . [ the InitS of M,[x,y]] = x and

A6: for x, y being Real st x >= y holds

the Tran of M . [ the InitS of M,[x,y]] = y ; :: thesis: for x, y being Element of REAL holds Result ([x,y],M) = min (x,y)

let x, y be Element of REAL ; :: thesis: Result ([x,y],M) = min (x,y)

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 Result ([x,y],M) = min (x,y) by A7, Def9; :: thesis: verum

the Tran of M . [ the InitS of M,[x,y]] = x ) & ( for x, y being Real st x >= y holds

the Tran of M . [ the InitS of M,[x,y]] = y ) implies for x, y being Element of REAL holds Result ([x,y],M) = min (x,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 and

A5: for x, y being Real st x < y holds

the Tran of M . [ the InitS of M,[x,y]] = x and

A6: for x, y being Real st x >= y holds

the Tran of M . [ the InitS of M,[x,y]] = y ; :: thesis: for x, y being Element of REAL holds Result ([x,y],M) = min (x,y)

let x, y be Element of REAL ; :: thesis: Result ([x,y],M) = min (x,y)

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 Result ([x,y],M) = min (x,y) by A7, Def9; :: thesis: verum