let n be Element of NAT ; :: thesis: for S being Gene-Set

for p1, p2 being Individual of S holds crossover (p1,p2,n,0) = crossover (p2,p1,n)

let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n,0) = crossover (p2,p1,n)

let p1, p2 be Individual of S; :: thesis: crossover (p1,p2,n,0) = crossover (p2,p1,n)

reconsider q1 = crossover (p1,p2,n) as Individual of S ;

reconsider q2 = crossover (p2,p1,n) as Individual of S ;

crossover (p1,p2,n,0) = crossover (q1,q2,0)

.= q2 by Th4 ;

hence crossover (p1,p2,n,0) = crossover (p2,p1,n) ; :: thesis: verum

for p1, p2 being Individual of S holds crossover (p1,p2,n,0) = crossover (p2,p1,n)

let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n,0) = crossover (p2,p1,n)

let p1, p2 be Individual of S; :: thesis: crossover (p1,p2,n,0) = crossover (p2,p1,n)

reconsider q1 = crossover (p1,p2,n) as Individual of S ;

reconsider q2 = crossover (p2,p1,n) as Individual of S ;

crossover (p1,p2,n,0) = crossover (q1,q2,0)

.= q2 by Th4 ;

hence crossover (p1,p2,n,0) = crossover (p2,p1,n) ; :: thesis: verum