let n1, n2, n3 be Element of NAT ; for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3)
let S be Gene-Set; for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3)
let p1, p2 be Individual of S; ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3) )
assume that
A1:
n1 >= len p1
and
A2:
n2 >= len p1
; crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3)
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3)
by A1, Th18;
hence
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3)
by A2, Th9; verum