let n1, n2 be Element of NAT ; for S being Gene-Set
for p1, p2 being Individual of S st n2 >= len p1 holds
crossover (p1,p2,n1,n2) = crossover (p1,p2,n1)
let S be Gene-Set; for p1, p2 being Individual of S st n2 >= len p1 holds
crossover (p1,p2,n1,n2) = crossover (p1,p2,n1)
let p1, p2 be Individual of S; ( n2 >= len p1 implies crossover (p1,p2,n1,n2) = crossover (p1,p2,n1) )
assume
n2 >= len p1
; crossover (p1,p2,n1,n2) = crossover (p1,p2,n1)
then
n2 >= len S
by Def1;
then A1:
n2 >= len (crossover (p1,p2,n1))
by Def1;
crossover (p1,p2,n1,n2) = crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n2)
;
hence
crossover (p1,p2,n1,n2) = crossover (p1,p2,n1)
by A1, Th5; verum