let n1, n2 be Element of NAT ; :: thesis: 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; :: thesis: 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; :: thesis: ( n2 >= len p1 implies crossover (p1,p2,n1,n2) = crossover (p1,p2,n1) )
assume n2 >= len p1 ; :: thesis: 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; :: thesis: verum