let n1, n3, n4, n5, n6, n2 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n1,n3,n4,n5,n6 = crossover p1,p2,n3,n4,n5,n6 & crossover p1,p2,n1,n2,n1,n4,n5,n6 = crossover p1,p2,n2,n4,n5,n6 & crossover p1,p2,n1,n2,n3,n1,n5,n6 = crossover p1,p2,n2,n3,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n1,n6 = crossover p1,p2,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n1 = crossover p1,p2,n2,n3,n4,n5 )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n1,n3,n4,n5,n6 = crossover p1,p2,n3,n4,n5,n6 & crossover p1,p2,n1,n2,n1,n4,n5,n6 = crossover p1,p2,n2,n4,n5,n6 & crossover p1,p2,n1,n2,n3,n1,n5,n6 = crossover p1,p2,n2,n3,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n1,n6 = crossover p1,p2,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n1 = crossover p1,p2,n2,n3,n4,n5 )
let p1, p2 be Individual of S; :: thesis: ( crossover p1,p2,n1,n1,n3,n4,n5,n6 = crossover p1,p2,n3,n4,n5,n6 & crossover p1,p2,n1,n2,n1,n4,n5,n6 = crossover p1,p2,n2,n4,n5,n6 & crossover p1,p2,n1,n2,n3,n1,n5,n6 = crossover p1,p2,n2,n3,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n1,n6 = crossover p1,p2,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n1 = crossover p1,p2,n2,n3,n4,n5 )
crossover p1,p2,n1,n1,n3,n4,n5,n6 = crossover (crossover p1,p2,n3,n4,n5),(crossover p2,p1,n1,n1,n3,n4,n5),n6 by Th52
.= crossover (crossover p1,p2,n3,n4,n5),(crossover p2,p1,n3,n4,n5),n6 by Th52 ;
hence crossover p1,p2,n1,n1,n3,n4,n5,n6 = crossover p1,p2,n3,n4,n5,n6 ; :: thesis: ( crossover p1,p2,n1,n2,n1,n4,n5,n6 = crossover p1,p2,n2,n4,n5,n6 & crossover p1,p2,n1,n2,n3,n1,n5,n6 = crossover p1,p2,n2,n3,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n1,n6 = crossover p1,p2,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n1 = crossover p1,p2,n2,n3,n4,n5 )
crossover p1,p2,n1,n2,n1,n4,n5,n6 = crossover (crossover p1,p2,n2,n4,n5),(crossover p2,p1,n1,n2,n1,n4,n5),n6 by Th52
.= crossover (crossover p1,p2,n2,n4,n5),(crossover p2,p1,n2,n4,n5),n6 by Th52 ;
hence crossover p1,p2,n1,n2,n1,n4,n5,n6 = crossover p1,p2,n2,n4,n5,n6 ; :: thesis: ( crossover p1,p2,n1,n2,n3,n1,n5,n6 = crossover p1,p2,n2,n3,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n1,n6 = crossover p1,p2,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n1 = crossover p1,p2,n2,n3,n4,n5 )
crossover p1,p2,n1,n2,n3,n1,n5,n6 = crossover (crossover p1,p2,n2,n3,n5),(crossover p2,p1,n1,n2,n3,n1,n5),n6 by Th52
.= crossover (crossover p1,p2,n2,n3,n5),(crossover p2,p1,n2,n3,n5),n6 by Th52 ;
hence crossover p1,p2,n1,n2,n3,n1,n5,n6 = crossover p1,p2,n2,n3,n5,n6 ; :: thesis: ( crossover p1,p2,n1,n2,n3,n4,n1,n6 = crossover p1,p2,n2,n3,n4,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n1 = crossover p1,p2,n2,n3,n4,n5 )
crossover p1,p2,n1,n2,n3,n4,n1,n6 = crossover (crossover p1,p2,n2,n3,n4),(crossover p2,p1,n1,n2,n3,n4,n1),n6 by Th52
.= crossover (crossover p1,p2,n2,n3,n4),(crossover p2,p1,n2,n3,n4),n6 by Th52 ;
hence crossover p1,p2,n1,n2,n3,n4,n1,n6 = crossover p1,p2,n2,n3,n4,n6 ; :: thesis: crossover p1,p2,n1,n2,n3,n4,n5,n1 = crossover p1,p2,n2,n3,n4,n5
crossover p1,p2,n1,n2,n3,n4,n5,n1 = crossover p1,p2,n5,n2,n3,n4,n1,n1 by Th56
.= crossover p1,p2,n1,n2,n3,n4,n1,n5 by Th56
.= crossover (crossover p1,p2,n2,n3,n4),(crossover p2,p1,n1,n2,n3,n4,n1),n5 by Th52
.= crossover (crossover p1,p2,n2,n3,n4),(crossover p2,p1,n2,n3,n4),n5 by Th52 ;
hence crossover p1,p2,n1,n2,n3,n4,n5,n1 = crossover p1,p2,n2,n3,n4,n5 ; :: thesis: verum