set A = [.0,1.];
deffunc H₁( Element of [.0,1.], Element of [.0,1.]) -> Element of [.0,1.] = In ((max ((1 - $1),(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 x, y being Element of [.0,1.] holds f . (x,y) = max ((1 - x),(min (x,y)))
let a, b be Element of [.0,1.]; :: thesis: f . (a,b) = max ((1 - a),(min (a,b)))
A2: f . (a,b) = H₁(a,b) by A1;
( max ((1 - a),(min (a,b))) = 1 - a or max ((1 - a),(min (a,b))) = min (a,b) ) by XXREAL_0:16;
hence f . (a,b) = max ((1 - a),(min (a,b))) by A2, SUBSET_1:def 8, FUZNORM1:7, FUZNORM1:1; :: thesis: verum