let n1, n2, n3, n4 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3,n4) = crossover (p2,p1,n2,n3,n4) & crossover (p1,p2,n1,0,n3,n4) = crossover (p2,p1,n1,n3,n4) & crossover (p1,p2,n1,n2,0,n4) = crossover (p2,p1,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3,n4) = crossover (p2,p1,n2,n3,n4) & crossover (p1,p2,n1,0,n3,n4) = crossover (p2,p1,n1,n3,n4) & crossover (p1,p2,n1,n2,0,n4) = crossover (p2,p1,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) )
let p1, p2 be Individual of S; :: thesis: ( crossover (p1,p2,0,n2,n3,n4) = crossover (p2,p1,n2,n3,n4) & crossover (p1,p2,n1,0,n3,n4) = crossover (p2,p1,n1,n3,n4) & crossover (p1,p2,n1,n2,0,n4) = crossover (p2,p1,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) )
crossover (p1,p2,0,n2,n3,n4) = crossover ((crossover (p2,p1,n2,n3)),(crossover (p2,p1,0,n2,n3)),n4) by Th15
.= crossover ((crossover (p2,p1,n2,n3)),(crossover (p1,p2,n2,n3)),n4) by Th15 ;
hence crossover (p1,p2,0,n2,n3,n4) = crossover (p2,p1,n2,n3,n4) ; :: thesis: ( crossover (p1,p2,n1,0,n3,n4) = crossover (p2,p1,n1,n3,n4) & crossover (p1,p2,n1,n2,0,n4) = crossover (p2,p1,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) )
crossover (p1,p2,n1,0,n3,n4) = crossover ((crossover (p2,p1,n1,n3)),(crossover (p2,p1,n1,0,n3)),n4) by Th15
.= crossover ((crossover (p2,p1,n1,n3)),(crossover (p1,p2,n1,n3)),n4) by Th15 ;
hence crossover (p1,p2,n1,0,n3,n4) = crossover (p2,p1,n1,n3,n4) ; :: thesis: ( crossover (p1,p2,n1,n2,0,n4) = crossover (p2,p1,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) )
crossover (p1,p2,n1,n2,0,n4) = crossover ((crossover (p2,p1,n1,n2)),(crossover (p2,p1,n1,n2,0)),n4) by Th15
.= crossover ((crossover (p2,p1,n1,n2)),(crossover (p1,p2,n1,n2)),n4) by Th15 ;
hence crossover (p1,p2,n1,n2,0,n4) = crossover (p2,p1,n1,n2,n4) ; :: thesis: crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3)
crossover (p1,p2,n1,n2,n3,0) = crossover ((crossover (p1,p2,n1,n2,n3)),(crossover (p2,p1,n1,n2,n3)),0) ;
hence crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) by Th4; :: thesis: verum