let n2, n1 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