let n1, n2, n3, n4, n5 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