let M, N be non empty symmetric MetrStruct ; :: thesis: max-Prod2 M,N is symmetric
let a, b be Element of (max-Prod2 M,N); :: according to METRIC_1:def 5,METRIC_1:def 9 :: thesis: the distance of (max-Prod2 M,N) . a,b = the distance of (max-Prod2 M,N) . b,a
consider x1, y1 being Point of M, x2, y2 being Point of N such that
A1: a = [x1,x2] and
A2: b = [y1,y2] and
A3: the distance of (max-Prod2 M,N) . a,b = max (the distance of M . x1,y1),(the distance of N . x2,y2) by Def1;
consider m1, n1 being Point of M, m2, n2 being Point of N such that
A4: b = [m1,m2] and
A5: a = [n1,n2] and
A6: the distance of (max-Prod2 M,N) . b,a = max (the distance of M . m1,n1),(the distance of N . m2,n2) by Def1;
A7: x1 = n1 by A1, A5, ZFMISC_1:33;
the distance of N is symmetric by METRIC_1:def 9;
then A8: the distance of N . x2,y2 = the distance of N . y2,x2 by METRIC_1:def 5;
the distance of M is symmetric by METRIC_1:def 9;
then A9: the distance of M . x1,y1 = the distance of M . y1,x1 by METRIC_1:def 5;
( y1 = m1 & y2 = m2 ) by A2, A4, ZFMISC_1:33;
hence the distance of (max-Prod2 M,N) . a,b = the distance of (max-Prod2 M,N) . b,a by A1, A3, A5, A6, A9, A8, A7, ZFMISC_1:33; :: thesis: verum