deffunc H₁( Real, Real) -> Element of REAL = In (($1 + $2),REAL);
consider f being BinOp of REAL such that
A1: for x, y being Element of REAL holds f . (x,y) = H₁(x,y) from BINOP_1:sch 4();
A2: for x, y being Element of REAL holds f . (x,y) = x + y

proof

let x, y be Element of REAL ; :: thesis: f . (x,y) = x + y
reconsider x = x, y = y as Real ;
f . (x,y) = H₁(x,y) by A1;
hence f . (x,y) = x + y ; :: thesis: verum

end;

let M be non empty Moore-SM_Final over [:REAL,REAL:], succ REAL; :: thesis: ( 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 being Real holds the Tran of M . [ the InitS of M,[x,y]] = x + y ) implies for x, y being Element of REAL holds x + y is_result_of [x,y],M )
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 ; :: thesis: ( ex x, y being Real st not the Tran of M . [ the InitS of M,[x,y]] = x + y or for x, y being Element of REAL holds x + y is_result_of [x,y],M )
assume A8: for x, y being Real holds the Tran of M . [ the InitS of M,[x,y]] = x + y ; :: thesis: for x, y being Element of REAL holds x + y is_result_of [x,y],M
let x, y be Element of REAL ; :: thesis: x + y is_result_of [x,y],M

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

let x, y be Element of REAL ; :: thesis: the Tran of M . [ the InitS of M,[x,y]] = f . (x,y)
the Tran of M . [ the InitS of M,[x,y]] = x + y by A8;
hence the Tran of M . [ the InitS of M,[x,y]] = f . (x,y) by A2; :: thesis: verum

end;

then f . (x,y) is_result_of [x,y],M by A3, A4, A5, A6, A7, Th22;
hence x + y is_result_of [x,y],M by A2; :: thesis: verum