let n3, n1, n2 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S st n3 >= len p1 holds
crossover p1,p2,n1,n2,n3 = crossover p1,p2,n1,n2
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S st n3 >= len p1 holds
crossover p1,p2,n1,n2,n3 = crossover p1,p2,n1,n2
let p1, p2 be Individual of S; :: thesis: ( n3 >= len p1 implies crossover p1,p2,n1,n2,n3 = crossover p1,p2,n1,n2 )
assume n3 >= len p1 ; :: thesis: crossover p1,p2,n1,n2,n3 = crossover p1,p2,n1,n2
then n3 >= len S by Def1;
then A1: n3 >= len (crossover p1,p2,n1,n2) by Def1;
crossover p1,p2,n1,n2,n3 = crossover (crossover p1,p2,n1,n2),(crossover p2,p1,n1,n2),n3 ;
hence crossover p1,p2,n1,n2,n3 = crossover p1,p2,n1,n2 by A1, Th5; :: thesis: verum