let n2, n3, n1 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,n2,n3 = crossover p2,p1,n2,n3 & crossover p1,p2,n1,0 ,n3 = crossover p2,p1,n1,n3 & crossover p1,p2,n1,n2,0 = crossover p2,p1,n1,n2 )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,n2,n3 = crossover p2,p1,n2,n3 & crossover p1,p2,n1,0 ,n3 = crossover p2,p1,n1,n3 & crossover p1,p2,n1,n2,0 = crossover p2,p1,n1,n2 )
let p1, p2 be Individual of S; :: thesis: ( crossover p1,p2,0 ,n2,n3 = crossover p2,p1,n2,n3 & crossover p1,p2,n1,0 ,n3 = crossover p2,p1,n1,n3 & crossover p1,p2,n1,n2,0 = crossover p2,p1,n1,n2 )
crossover p1,p2,0 ,n2,n3 = crossover (crossover p2,p1,n2),(crossover p2,p1,0 ,n2),n3 by Th7
.= crossover (crossover p2,p1,n2),(crossover p1,p2,n2),n3 by Th7 ;
hence crossover p1,p2,0 ,n2,n3 = crossover p2,p1,n2,n3 ; :: thesis: ( crossover p1,p2,n1,0 ,n3 = crossover p2,p1,n1,n3 & crossover p1,p2,n1,n2,0 = crossover p2,p1,n1,n2 )
crossover p1,p2,n1,0 ,n3 = crossover (crossover p2,p1,n1),(crossover p2,p1,n1,0 ),n3 by Th8
.= crossover (crossover p2,p1,n1),(crossover p1,p2,n1),n3 by Th8 ;
hence crossover p1,p2,n1,0 ,n3 = crossover p2,p1,n1,n3 ; :: thesis: crossover p1,p2,n1,n2,0 = crossover p2,p1,n1,n2
crossover p1,p2,n1,n2,0 = crossover (crossover p1,p2,n1,n2),(crossover p2,p1,n1,n2),0 ;
hence crossover p1,p2,n1,n2,0 = crossover p2,p1,n1,n2 by Th4; :: thesis: verum