set A = [.0,1.];

deffunc H_{1}( Element of [.0,1.], Element of [.0,1.]) -> Element of [.0,1.] = In ((min ($1,$2)),[.0,1.]);

ex f being Function of [:[.0,1.],[.0,1.]:],[.0,1.] st

for x, y being Element of [.0,1.] holds f . (x,y) = H_{1}(x,y)
from BINOP_1:sch 4();

then consider f being BinOp of [.0,1.] such that

A1: for x, y being Element of [.0,1.] holds f . (x,y) = H_{1}(x,y)
;

take f ; :: thesis: for a, b being Element of [.0,1.] holds f . (a,b) = min (a,b)

let a, b be Element of [.0,1.]; :: thesis: f . (a,b) = min (a,b)

A2: f . (a,b) = H_{1}(a,b)
by A1;

( min (a,b) = a or min (a,b) = b ) by XXREAL_0:15;

hence f . (a,b) = min (a,b) by A2, SUBSET_1:def 8; :: thesis: verum

deffunc H

ex f being Function of [:[.0,1.],[.0,1.]:],[.0,1.] st

for x, y being Element of [.0,1.] holds f . (x,y) = H

then consider f being BinOp of [.0,1.] such that

A1: for x, y being Element of [.0,1.] holds f . (x,y) = H

take f ; :: thesis: for a, b being Element of [.0,1.] holds f . (a,b) = min (a,b)

let a, b be Element of [.0,1.]; :: thesis: f . (a,b) = min (a,b)

A2: f . (a,b) = H

( min (a,b) = a or min (a,b) = b ) by XXREAL_0:15;

hence f . (a,b) = min (a,b) by A2, SUBSET_1:def 8; :: thesis: verum