let n1, n2, n3 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3)
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S st n1 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3)
let p1, p2 be Individual of S; :: thesis: ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3) )
assume A1: n1 >= len p1 ; :: thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3)
then n1 >= len S by Def1;
then A2: n1 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3) = crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n1,n2)),n3) by A1, Th9
.= crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n2)),n3) by A2, Th9 ;
hence crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3) ; :: thesis: verum