let n1, n2, n3 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n1,n3) & crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2) )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n1,n3) & crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2) )
let p1, p2 be Individual of S; :: thesis: ( crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n1,n3) & crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2) )
set q1 = crossover (p1,p2,n1);
set q2 = crossover (p2,p1,n1);
crossover (p1,p2,n1,n2,n3) = crossover ((crossover (p1,p2,n2,n1)),(crossover (p2,p1,n1,n2)),n3) by Th13
.= crossover ((crossover (p1,p2,n2,n1)),(crossover (p2,p1,n2,n1)),n3) by Th13 ;
hence crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n1,n3) ; :: thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2)
crossover (p1,p2,n1,n2,n3) = crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n2,n3)
.= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n3,n2) by Th13
.= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n2) ;
hence crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2) ; :: thesis: verum