let M, N be non empty MetrStruct ; :: thesis: for m1, m2 being Point of M

for n1, n2 being Point of N holds dist ([m1,n1],[m2,n2]) = max ((dist (m1,m2)),(dist (n1,n2)))

let m1, m2 be Point of M; :: thesis: for n1, n2 being Point of N holds dist ([m1,n1],[m2,n2]) = max ((dist (m1,m2)),(dist (n1,n2)))

let n1, n2 be Point of N; :: thesis: dist ([m1,n1],[m2,n2]) = max ((dist (m1,m2)),(dist (n1,n2)))

consider x1, y1 being Point of M, x2, y2 being Point of N such that

A1: [m1,n1] = [x1,x2] and

A2: [m2,n2] = [y1,y2] and

A3: the distance of (max-Prod2 (M,N)) . ([m1,n1],[m2,n2]) = max (( the distance of M . (x1,y1)),( the distance of N . (x2,y2))) by Def1;

A4: m2 = y1 by A2, XTUPLE_0:1;

( m1 = x1 & n1 = x2 ) by A1, XTUPLE_0:1;

hence dist ([m1,n1],[m2,n2]) = max ((dist (m1,m2)),(dist (n1,n2))) by A2, A3, A4, XTUPLE_0:1; :: thesis: verum

for n1, n2 being Point of N holds dist ([m1,n1],[m2,n2]) = max ((dist (m1,m2)),(dist (n1,n2)))

let m1, m2 be Point of M; :: thesis: for n1, n2 being Point of N holds dist ([m1,n1],[m2,n2]) = max ((dist (m1,m2)),(dist (n1,n2)))

let n1, n2 be Point of N; :: thesis: dist ([m1,n1],[m2,n2]) = max ((dist (m1,m2)),(dist (n1,n2)))

consider x1, y1 being Point of M, x2, y2 being Point of N such that

A1: [m1,n1] = [x1,x2] and

A2: [m2,n2] = [y1,y2] and

A3: the distance of (max-Prod2 (M,N)) . ([m1,n1],[m2,n2]) = max (( the distance of M . (x1,y1)),( the distance of N . (x2,y2))) by Def1;

A4: m2 = y1 by A2, XTUPLE_0:1;

( m1 = x1 & n1 = x2 ) by A1, XTUPLE_0:1;

hence dist ([m1,n1],[m2,n2]) = max ((dist (m1,m2)),(dist (n1,n2))) by A2, A3, A4, XTUPLE_0:1; :: thesis: verum