let M, N be non empty Reflexive MetrStruct ; :: thesis: max-Prod2 (M,N) is Reflexive
let a be Element of (max-Prod2 (M,N)); :: according to METRIC_1:def 2,METRIC_1:def 6 :: thesis: the distance of (max-Prod2 (M,N)) . (a,a) = 0
consider x1, y1 being Point of M, x2, y2 being Point of N such that
A1: ( a = [x1,x2] & a = [y1,y2] ) and
A2: the distance of (max-Prod2 (M,N)) . (a,a) = max (( the distance of M . (x1,y1)),( the distance of N . (x2,y2))) by Def1;
the distance of M is Reflexive by METRIC_1:def 6;
then A3: the distance of M . (x1,x1) = 0 ;
the distance of N is Reflexive by METRIC_1:def 6;
then A4: the distance of N . (x2,x2) = 0 ;
( x1 = y1 & x2 = y2 ) by A1, XTUPLE_0:1;
hence the distance of (max-Prod2 (M,N)) . (a,a) = 0 by A2, A3, A4; :: thesis: verum