let n5, n4, n3, n2, n1 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,0 ,0 ,0 ,n5 = crossover p1,p2,n5 & crossover p1,p2,0 ,0 ,0 ,n4,0 = crossover p1,p2,n4 & crossover p1,p2,0 ,0 ,n3,0 ,0 = crossover p1,p2,n3 & crossover p1,p2,0 ,n2,0 ,0 ,0 = crossover p1,p2,n2 & crossover p1,p2,n1,0 ,0 ,0 ,0 = crossover p1,p2,n1 )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,0 ,0 ,0 ,n5 = crossover p1,p2,n5 & crossover p1,p2,0 ,0 ,0 ,n4,0 = crossover p1,p2,n4 & crossover p1,p2,0 ,0 ,n3,0 ,0 = crossover p1,p2,n3 & crossover p1,p2,0 ,n2,0 ,0 ,0 = crossover p1,p2,n2 & crossover p1,p2,n1,0 ,0 ,0 ,0 = crossover p1,p2,n1 )
let p1, p2 be Individual of S; :: thesis: ( crossover p1,p2,0 ,0 ,0 ,0 ,n5 = crossover p1,p2,n5 & crossover p1,p2,0 ,0 ,0 ,n4,0 = crossover p1,p2,n4 & crossover p1,p2,0 ,0 ,n3,0 ,0 = crossover p1,p2,n3 & crossover p1,p2,0 ,n2,0 ,0 ,0 = crossover p1,p2,n2 & crossover p1,p2,n1,0 ,0 ,0 ,0 = crossover p1,p2,n1 )
A1: ( crossover p1,p2,0 ,0 ,n3,0 ,0 = crossover (crossover p1,p2,0 ,0 ,n3,0 ),(crossover p1,p2,n3),0 & crossover p1,p2,0 ,n2,0 ,0 ,0 = crossover (crossover p1,p2,0 ,n2,0 ,0 ),(crossover p1,p2,n2),0 ) by Th31;
A2: crossover p1,p2,n1,0 ,0 ,0 ,0 = crossover (crossover p1,p2,n1,0 ,0 ,0 ),(crossover p1,p2,n1),0 by Th31;
( crossover p1,p2,0 ,0 ,0 ,0 ,n5 = crossover p1,(crossover p2,p1,0 ,0 ,0 ,0 ),n5 & crossover p1,p2,0 ,0 ,0 ,n4,0 = crossover (crossover p1,p2,0 ,0 ,0 ,n4),(crossover p1,p2,n4),0 ) by Th31, Th32;
hence ( crossover p1,p2,0 ,0 ,0 ,0 ,n5 = crossover p1,p2,n5 & crossover p1,p2,0 ,0 ,0 ,n4,0 = crossover p1,p2,n4 & crossover p1,p2,0 ,0 ,n3,0 ,0 = crossover p1,p2,n3 & crossover p1,p2,0 ,n2,0 ,0 ,0 = crossover p1,p2,n2 & crossover p1,p2,n1,0 ,0 ,0 ,0 = crossover p1,p2,n1 ) by A1, A2, Th4, Th32; :: thesis: verum