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,n2,n3,n4,n5) = crossover (p2,p1,n2,n3,n4,n5) & crossover (p1,p2,n1,0,n3,n4,n5) = crossover (p2,p1,n1,n3,n4,n5) & crossover (p1,p2,n1,n2,0,n4,n5) = crossover (p2,p1,n1,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,0,n5) = crossover (p2,p1,n1,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,0) = crossover (p2,p1,n1,n2,n3,n4) )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3,n4,n5) = crossover (p2,p1,n2,n3,n4,n5) & crossover (p1,p2,n1,0,n3,n4,n5) = crossover (p2,p1,n1,n3,n4,n5) & crossover (p1,p2,n1,n2,0,n4,n5) = crossover (p2,p1,n1,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,0,n5) = crossover (p2,p1,n1,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,0) = crossover (p2,p1,n1,n2,n3,n4) )
let p1, p2 be Individual of S; :: thesis: ( crossover (p1,p2,0,n2,n3,n4,n5) = crossover (p2,p1,n2,n3,n4,n5) & crossover (p1,p2,n1,0,n3,n4,n5) = crossover (p2,p1,n1,n3,n4,n5) & crossover (p1,p2,n1,n2,0,n4,n5) = crossover (p2,p1,n1,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,0,n5) = crossover (p2,p1,n1,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,0) = crossover (p2,p1,n1,n2,n3,n4) )
A1: crossover (p1,p2,n1,n2,n3,n4,0) = crossover ((crossover (p1,p2,n1,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),0) ;
A2: crossover (p1,p2,n1,0,n3,n4,n5) = crossover ((crossover (p2,p1,n1,n3,n4)),(crossover (p2,p1,n1,0,n3,n4)),n5) by Th29
.= crossover ((crossover (p2,p1,n1,n3,n4)),(crossover (p1,p2,n1,n3,n4)),n5) by Th29 ;
A3: crossover (p1,p2,n1,n2,n3,0,n5) = crossover ((crossover (p2,p1,n1,n2,n3)),(crossover (p2,p1,n1,n2,n3,0)),n5) by Th29
.= crossover ((crossover (p2,p1,n1,n2,n3)),(crossover (p1,p2,n1,n2,n3)),n5) by Th29 ;
A4: crossover (p1,p2,n1,n2,0,n4,n5) = crossover ((crossover (p2,p1,n1,n2,n4)),(crossover (p2,p1,n1,n2,0,n4)),n5) by Th29
.= crossover ((crossover (p2,p1,n1,n2,n4)),(crossover (p1,p2,n1,n2,n4)),n5) by Th29 ;
crossover (p1,p2,0,n2,n3,n4,n5) = crossover ((crossover (p2,p1,n2,n3,n4)),(crossover (p2,p1,0,n2,n3,n4)),n5) by Th29
.= crossover ((crossover (p2,p1,n2,n3,n4)),(crossover (p1,p2,n2,n3,n4)),n5) by Th29 ;
hence ( crossover (p1,p2,0,n2,n3,n4,n5) = crossover (p2,p1,n2,n3,n4,n5) & crossover (p1,p2,n1,0,n3,n4,n5) = crossover (p2,p1,n1,n3,n4,n5) & crossover (p1,p2,n1,n2,0,n4,n5) = crossover (p2,p1,n1,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,0,n5) = crossover (p2,p1,n1,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,0) = crossover (p2,p1,n1,n2,n3,n4) ) by A2, A4, A3, A1, Th4; :: thesis: verum