let n4, n5, n3, n2, n1 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