set A = [.0,1.];
deffunc H₁( 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) = H₁(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) = H₁(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) = H₁(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