let n2, n3, n4, n5, n6, 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,n4,n5,n6 = crossover p2,p1,n2,n3,n4,n5,n6 & crossover p1,p2,n1,0 ,n3,n4,n5,n6 = crossover p2,p1,n1,n3,n4,n5,n6 & crossover p1,p2,n1,n2,0 ,n4,n5,n6 = crossover p2,p1,n1,n2,n4,n5,n6 & crossover p1,p2,n1,n2,n3,0 ,n5,n6 = crossover p2,p1,n1,n2,n3,n5,n6 & crossover p1,p2,n1,n2,n3,n4,0 ,n6 = crossover p2,p1,n1,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,0 = crossover p2,p1,n1,n2,n3,n4,n5 )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,n2,n3,n4,n5,n6 = crossover p2,p1,n2,n3,n4,n5,n6 & crossover p1,p2,n1,0 ,n3,n4,n5,n6 = crossover p2,p1,n1,n3,n4,n5,n6 & crossover p1,p2,n1,n2,0 ,n4,n5,n6 = crossover p2,p1,n1,n2,n4,n5,n6 & crossover p1,p2,n1,n2,n3,0 ,n5,n6 = crossover p2,p1,n1,n2,n3,n5,n6 & crossover p1,p2,n1,n2,n3,n4,0 ,n6 = crossover p2,p1,n1,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,0 = crossover p2,p1,n1,n2,n3,n4,n5 )
let p1, p2 be Individual of S; :: thesis: ( crossover p1,p2,0 ,n2,n3,n4,n5,n6 = crossover p2,p1,n2,n3,n4,n5,n6 & crossover p1,p2,n1,0 ,n3,n4,n5,n6 = crossover p2,p1,n1,n3,n4,n5,n6 & crossover p1,p2,n1,n2,0 ,n4,n5,n6 = crossover p2,p1,n1,n2,n4,n5,n6 & crossover p1,p2,n1,n2,n3,0 ,n5,n6 = crossover p2,p1,n1,n2,n3,n5,n6 & crossover p1,p2,n1,n2,n3,n4,0 ,n6 = crossover p2,p1,n1,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,0 = crossover p2,p1,n1,n2,n3,n4,n5 )
crossover p1,p2,0 ,n2,n3,n4,n5,n6 = crossover (crossover p2,p1,n2,n3,n4,n5),(crossover p2,p1,0 ,n2,n3,n4,n5),n6 by Th41
.= crossover (crossover p2,p1,n2,n3,n4,n5),(crossover p1,p2,n2,n3,n4,n5),n6 by Th41 ;
hence crossover p1,p2,0 ,n2,n3,n4,n5,n6 = crossover p2,p1,n2,n3,n4,n5,n6 ; :: thesis: ( crossover p1,p2,n1,0 ,n3,n4,n5,n6 = crossover p2,p1,n1,n3,n4,n5,n6 & crossover p1,p2,n1,n2,0 ,n4,n5,n6 = crossover p2,p1,n1,n2,n4,n5,n6 & crossover p1,p2,n1,n2,n3,0 ,n5,n6 = crossover p2,p1,n1,n2,n3,n5,n6 & crossover p1,p2,n1,n2,n3,n4,0 ,n6 = crossover p2,p1,n1,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,0 = crossover p2,p1,n1,n2,n3,n4,n5 )
crossover p1,p2,n1,0 ,n3,n4,n5,n6 = crossover (crossover p2,p1,n1,n3,n4,n5),(crossover p2,p1,n1,0 ,n3,n4,n5),n6 by Th41
.= crossover (crossover p2,p1,n1,n3,n4,n5),(crossover p1,p2,n1,n3,n4,n5),n6 by Th41 ;
hence crossover p1,p2,n1,0 ,n3,n4,n5,n6 = crossover p2,p1,n1,n3,n4,n5,n6 ; :: thesis: ( crossover p1,p2,n1,n2,0 ,n4,n5,n6 = crossover p2,p1,n1,n2,n4,n5,n6 & crossover p1,p2,n1,n2,n3,0 ,n5,n6 = crossover p2,p1,n1,n2,n3,n5,n6 & crossover p1,p2,n1,n2,n3,n4,0 ,n6 = crossover p2,p1,n1,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,0 = crossover p2,p1,n1,n2,n3,n4,n5 )
crossover p1,p2,n1,n2,0 ,n4,n5,n6 = crossover (crossover p2,p1,n1,n2,n4,n5),(crossover p2,p1,n1,n2,0 ,n4,n5),n6 by Th41
.= crossover (crossover p2,p1,n1,n2,n4,n5),(crossover p1,p2,n1,n2,n4,n5),n6 by Th41 ;
hence crossover p1,p2,n1,n2,0 ,n4,n5,n6 = crossover p2,p1,n1,n2,n4,n5,n6 ; :: thesis: ( crossover p1,p2,n1,n2,n3,0 ,n5,n6 = crossover p2,p1,n1,n2,n3,n5,n6 & crossover p1,p2,n1,n2,n3,n4,0 ,n6 = crossover p2,p1,n1,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,0 = crossover p2,p1,n1,n2,n3,n4,n5 )
crossover p1,p2,n1,n2,n3,0 ,n5,n6 = crossover (crossover p2,p1,n1,n2,n3,n5),(crossover p2,p1,n1,n2,n3,0 ,n5),n6 by Th41
.= crossover (crossover p2,p1,n1,n2,n3,n5),(crossover p1,p2,n1,n2,n3,n5),n6 by Th41 ;
hence crossover p1,p2,n1,n2,n3,0 ,n5,n6 = crossover p2,p1,n1,n2,n3,n5,n6 ; :: thesis: ( crossover p1,p2,n1,n2,n3,n4,0 ,n6 = crossover p2,p1,n1,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,0 = crossover p2,p1,n1,n2,n3,n4,n5 )
crossover p1,p2,n1,n2,n3,n4,0 ,n6 = crossover (crossover p2,p1,n1,n2,n3,n4),(crossover p2,p1,n1,n2,n3,n4,0 ),n6 by Th41
.= crossover (crossover p2,p1,n1,n2,n3,n4),(crossover p1,p2,n1,n2,n3,n4),n6 by Th41 ;
hence crossover p1,p2,n1,n2,n3,n4,0 ,n6 = crossover p2,p1,n1,n2,n3,n4,n6 ; :: thesis: crossover p1,p2,n1,n2,n3,n4,n5,0 = crossover p2,p1,n1,n2,n3,n4,n5
thus crossover p1,p2,n1,n2,n3,n4,n5,0 = crossover (crossover p1,p2,n1,n2,n3,n4,n5),(crossover p2,p1,n1,n2,n3,n4,n5),0
.= crossover p2,p1,n1,n2,n3,n4,n5 by Th4 ; :: thesis: verum