let n1, n2, n3, n4, n5, n6 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n1,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n3,n2,n1,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n4,n2,n3,n1,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n5,n2,n3,n4,n1,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n6,n2,n3,n4,n5,n1) )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n1,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n3,n2,n1,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n4,n2,n3,n1,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n5,n2,n3,n4,n1,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n6,n2,n3,n4,n5,n1) )
let p1, p2 be Individual of S; :: thesis: ( crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n1,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n3,n2,n1,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n4,n2,n3,n1,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n5,n2,n3,n4,n1,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n6,n2,n3,n4,n5,n1) )
set q1 = crossover (p1,p2,n5,n2,n3,n4);
set q2 = crossover (p2,p1,n5,n2,n3,n4);
A1: crossover (p1,p2,n5,n2,n3,n4,n6) = crossover ((crossover (p1,p2,n5,n2,n3,n4)),(crossover (p2,p1,n5,n2,n3,n4)),n6) ;
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n2,n1,n3,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by Th51
.= crossover ((crossover (p1,p2,n2,n1,n3,n4,n5)),(crossover (p2,p1,n2,n1,n3,n4,n5)),n6) by Th51 ;
hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n1,n3,n4,n5,n6) ; :: thesis: ( crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n3,n2,n1,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n4,n2,n3,n1,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n5,n2,n3,n4,n1,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n6,n2,n3,n4,n5,n1) )
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n3,n2,n1,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by Th51
.= crossover ((crossover (p1,p2,n3,n2,n1,n4,n5)),(crossover (p2,p1,n3,n2,n1,n4,n5)),n6) by Th51 ;
hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n3,n2,n1,n4,n5,n6) ; :: thesis: ( crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n4,n2,n3,n1,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n5,n2,n3,n4,n1,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n6,n2,n3,n4,n5,n1) )
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n4,n2,n3,n1,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by Th51
.= crossover ((crossover (p1,p2,n4,n2,n3,n1,n5)),(crossover (p2,p1,n4,n2,n3,n1,n5)),n6) by Th51 ;
hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n4,n2,n3,n1,n5,n6) ; :: thesis: ( crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n5,n2,n3,n4,n1,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n6,n2,n3,n4,n5,n1) )
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n5,n2,n3,n4,n1)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by Th51
.= crossover ((crossover (p1,p2,n5,n2,n3,n4,n1)),(crossover (p2,p1,n5,n2,n3,n4,n1)),n6) by Th51 ;
hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n5,n2,n3,n4,n1,n6) ; :: thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n6,n2,n3,n4,n5,n1)
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n5,n2,n3,n4,n1)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by Th51
.= crossover ((crossover (p1,p2,n5,n2,n3,n4,n1)),(crossover (p2,p1,n5,n2,n3,n4,n1)),n6) by Th51
.= crossover ((crossover (p1,p2,n5,n2,n3,n4)),(crossover (p2,p1,n5,n2,n3,n4)),n1,n6)
.= crossover ((crossover (p1,p2,n5,n2,n3,n4)),(crossover (p2,p1,n5,n2,n3,n4)),n6,n1) by Th13
.= crossover ((crossover (p1,p2,n6,n2,n3,n4,n5)),(crossover (p2,p1,n5,n2,n3,n4,n6)),n1) by A1, Th51
.= crossover ((crossover (p1,p2,n6,n2,n3,n4,n5)),(crossover (p2,p1,n6,n2,n3,n4,n5)),n1) by Th51 ;
hence crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n6,n2,n3,n4,n5,n1) ; :: thesis: verum