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,p2,0,0,0,n4,n5) = crossover (p2,p1,n4,n5) & crossover (p1,p2,0,0,n3,0,n5) = crossover (p2,p1,n3,n5) & crossover (p1,p2,0,0,n3,n4,0) = crossover (p2,p1,n3,n4) & crossover (p1,p2,0,n2,0,0,n5) = crossover (p2,p1,n2,n5) & crossover (p1,p2,0,n2,0,n4,0) = crossover (p2,p1,n2,n4) & crossover (p1,p2,0,n2,n3,0,0) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,0,0,n5) = crossover (p2,p1,n1,n5) & crossover (p1,p2,n1,0,0,n4,0) = crossover (p2,p1,n1,n4) & crossover (p1,p2,n1,0,n3,0,0) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0,0,0) = crossover (p2,p1,n1,n2) )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,0,n4,n5) = crossover (p2,p1,n4,n5) & crossover (p1,p2,0,0,n3,0,n5) = crossover (p2,p1,n3,n5) & crossover (p1,p2,0,0,n3,n4,0) = crossover (p2,p1,n3,n4) & crossover (p1,p2,0,n2,0,0,n5) = crossover (p2,p1,n2,n5) & crossover (p1,p2,0,n2,0,n4,0) = crossover (p2,p1,n2,n4) & crossover (p1,p2,0,n2,n3,0,0) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,0,0,n5) = crossover (p2,p1,n1,n5) & crossover (p1,p2,n1,0,0,n4,0) = crossover (p2,p1,n1,n4) & crossover (p1,p2,n1,0,n3,0,0) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0,0,0) = crossover (p2,p1,n1,n2) )
let p1, p2 be Individual of S; :: thesis: ( crossover (p1,p2,0,0,0,n4,n5) = crossover (p2,p1,n4,n5) & crossover (p1,p2,0,0,n3,0,n5) = crossover (p2,p1,n3,n5) & crossover (p1,p2,0,0,n3,n4,0) = crossover (p2,p1,n3,n4) & crossover (p1,p2,0,n2,0,0,n5) = crossover (p2,p1,n2,n5) & crossover (p1,p2,0,n2,0,n4,0) = crossover (p2,p1,n2,n4) & crossover (p1,p2,0,n2,n3,0,0) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,0,0,n5) = crossover (p2,p1,n1,n5) & crossover (p1,p2,n1,0,0,n4,0) = crossover (p2,p1,n1,n4) & crossover (p1,p2,n1,0,n3,0,0) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0,0,0) = crossover (p2,p1,n1,n2) )
A1: ( crossover (p1,p2,0,0,n3,n4,0) = crossover ((crossover (p1,p2,0,0,n3,n4)),(crossover (p2,p1,n3,n4)),0) & crossover (p1,p2,0,n2,0,n4,0) = crossover ((crossover (p1,p2,0,n2,0,n4)),(crossover (p2,p1,n2,n4)),0) ) by Th30;
A2: ( crossover (p1,p2,0,n2,n3,0,0) = crossover ((crossover (p1,p2,0,n2,n3,0)),(crossover (p2,p1,n2,n3)),0) & crossover (p1,p2,n1,0,0,n4,0) = crossover ((crossover (p1,p2,n1,0,0,n4)),(crossover (p2,p1,n1,n4)),0) ) by Th30;
A3: crossover (p1,p2,0,0,n3,0,n5) = crossover ((crossover (p2,p1,n3)),(crossover (p2,p1,0,0,n3,0)),n5) by Th31
.= crossover ((crossover (p2,p1,n3)),(crossover (p1,p2,n3)),n5) by Th31 ;
A4: ( crossover (p1,p2,n1,0,n3,0,0) = crossover ((crossover (p1,p2,n1,0,n3,0)),(crossover (p2,p1,n1,n3)),0) & crossover (p1,p2,n1,n2,0,0,0) = crossover ((crossover (p1,p2,n1,n2,0,0)),(crossover (p2,p1,n1,n2)),0) ) by Th30;
A5: crossover (p1,p2,n1,0,0,0,n5) = crossover ((crossover (p2,p1,n1)),(crossover (p2,p1,n1,0,0,0)),n5) by Th31
.= crossover ((crossover (p2,p1,n1)),(crossover (p1,p2,n1)),n5) by Th31 ;
A6: crossover (p1,p2,0,n2,0,0,n5) = crossover ((crossover (p2,p1,n2)),(crossover (p2,p1,0,n2,0,0)),n5) by Th31
.= crossover ((crossover (p2,p1,n2)),(crossover (p1,p2,n2)),n5) by Th31 ;
crossover (p1,p2,0,0,0,n4,n5) = crossover ((crossover (p2,p1,n4)),(crossover (p2,p1,0,0,0,n4)),n5) by Th31
.= crossover ((crossover (p2,p1,n4)),(crossover (p1,p2,n4)),n5) by Th31 ;
hence ( crossover (p1,p2,0,0,0,n4,n5) = crossover (p2,p1,n4,n5) & crossover (p1,p2,0,0,n3,0,n5) = crossover (p2,p1,n3,n5) & crossover (p1,p2,0,0,n3,n4,0) = crossover (p2,p1,n3,n4) & crossover (p1,p2,0,n2,0,0,n5) = crossover (p2,p1,n2,n5) & crossover (p1,p2,0,n2,0,n4,0) = crossover (p2,p1,n2,n4) & crossover (p1,p2,0,n2,n3,0,0) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,0,0,n5) = crossover (p2,p1,n1,n5) & crossover (p1,p2,n1,0,0,n4,0) = crossover (p2,p1,n1,n4) & crossover (p1,p2,n1,0,n3,0,0) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0,0,0) = crossover (p2,p1,n1,n2) ) by A3, A6, A1, A5, A2, A4, Th4; :: thesis: verum