let n1, n2 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) = crossover (p1,p2,n2)
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S st n1 >= len p1 holds
crossover (p1,p2,n1,n2) = crossover (p1,p2,n2)
let p1, p2 be Individual of S; :: thesis: ( n1 >= len p1 implies crossover (p1,p2,n1,n2) = crossover (p1,p2,n2) )
assume A1: n1 >= len p1 ; :: thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2)
then n1 >= len S by Def1;
then A2: n1 >= len p2 by Def1;
crossover (p1,p2,n1,n2) = crossover (p1,(crossover (p2,p1,n1)),n2) by A1, Th5;
hence crossover (p1,p2,n1,n2) = crossover (p1,p2,n2) by A2, Th5; :: thesis: verum