let n1, n2, n3, n4, n5 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) = crossover (p1,p2,n3,n4,n5) & crossover (p1,p2,n1,n2,n1,n4,n5) = crossover (p1,p2,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,n1,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) & crossover (p1,p2,n1,n2,n1,n4,n5) = crossover (p1,p2,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,n1,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) )
let p1, p2 be Individual of S; :: thesis: ( crossover (p1,p2,n1,n1,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) & crossover (p1,p2,n1,n2,n1,n4,n5) = crossover (p1,p2,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,n1,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) )
set q1 = crossover (p1,p2,n2,n3,n4);
set q2 = crossover (p2,p1,n2,n3,n4);
crossover (p1,p2,n1,n1,n3,n4,n5) = crossover ((crossover (p1,p2,n3,n4)),(crossover (p2,p1,n1,n1,n3,n4)),n5) by Th38
.= crossover ((crossover (p1,p2,n3,n4)),(crossover (p2,p1,n3,n4)),n5) by Th38 ;
hence crossover (p1,p2,n1,n1,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) ; :: thesis: ( crossover (p1,p2,n1,n2,n1,n4,n5) = crossover (p1,p2,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,n1,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) )
crossover (p1,p2,n1,n2,n1,n4,n5) = crossover ((crossover (p1,p2,n2,n4)),(crossover (p2,p1,n1,n2,n1,n4)),n5) by Th38
.= crossover ((crossover (p1,p2,n2,n4)),(crossover (p2,p1,n2,n4)),n5) by Th38 ;
hence crossover (p1,p2,n1,n2,n1,n4,n5) = crossover (p1,p2,n2,n4,n5) ; :: thesis: ( crossover (p1,p2,n1,n2,n3,n1,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) )
crossover (p1,p2,n1,n2,n3,n1,n5) = crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n1,n2,n3,n1)),n5) by Th38
.= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n5) by Th38 ;
hence crossover (p1,p2,n1,n2,n3,n1,n5) = crossover (p1,p2,n2,n3,n5) ; :: thesis: crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4)
crossover (p1,p2,n1,n2,n3,n4,n1) = crossover ((crossover (p1,p2,n2,n3,n4,n1)),(crossover (p2,p1,n1,n2,n3,n4)),n1) by Th37
.= crossover ((crossover (p1,p2,n2,n3,n4,n1)),(crossover (p2,p1,n2,n3,n4,n1)),n1) by Th37
.= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n1,n1) ;
hence crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) by Th12; :: thesis: verum