let n2, n3, n4, n5, n1 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