let n2, n3, n4, 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 = crossover p2,p1,n2,n3,n4 & crossover p1,p2,n1,0 ,n3,n4 = crossover p2,p1,n1,n3,n4 & crossover p1,p2,n1,n2,0 ,n4 = crossover p2,p1,n1,n2,n4 & crossover p1,p2,n1,n2,n3,0 = crossover p2,p1,n1,n2,n3 )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover p1,p2,0 ,n2,n3,n4 = crossover p2,p1,n2,n3,n4 & crossover p1,p2,n1,0 ,n3,n4 = crossover p2,p1,n1,n3,n4 & crossover p1,p2,n1,n2,0 ,n4 = crossover p2,p1,n1,n2,n4 & crossover p1,p2,n1,n2,n3,0 = crossover p2,p1,n1,n2,n3 )
let p1, p2 be Individual of S; :: thesis: ( crossover p1,p2,0 ,n2,n3,n4 = crossover p2,p1,n2,n3,n4 & crossover p1,p2,n1,0 ,n3,n4 = crossover p2,p1,n1,n3,n4 & crossover p1,p2,n1,n2,0 ,n4 = crossover p2,p1,n1,n2,n4 & crossover p1,p2,n1,n2,n3,0 = crossover p2,p1,n1,n2,n3 )
crossover p1,p2,0 ,n2,n3,n4 = crossover (crossover p2,p1,n2,n3),(crossover p2,p1,0 ,n2,n3),n4 by Th15
.= crossover (crossover p2,p1,n2,n3),(crossover p1,p2,n2,n3),n4 by Th15 ;
hence crossover p1,p2,0 ,n2,n3,n4 = crossover p2,p1,n2,n3,n4 ; :: thesis: ( crossover p1,p2,n1,0 ,n3,n4 = crossover p2,p1,n1,n3,n4 & crossover p1,p2,n1,n2,0 ,n4 = crossover p2,p1,n1,n2,n4 & crossover p1,p2,n1,n2,n3,0 = crossover p2,p1,n1,n2,n3 )
crossover p1,p2,n1,0 ,n3,n4 = crossover (crossover p2,p1,n1,n3),(crossover p2,p1,n1,0 ,n3),n4 by Th15
.= crossover (crossover p2,p1,n1,n3),(crossover p1,p2,n1,n3),n4 by Th15 ;
hence crossover p1,p2,n1,0 ,n3,n4 = crossover p2,p1,n1,n3,n4 ; :: thesis: ( crossover p1,p2,n1,n2,0 ,n4 = crossover p2,p1,n1,n2,n4 & crossover p1,p2,n1,n2,n3,0 = crossover p2,p1,n1,n2,n3 )
crossover p1,p2,n1,n2,0 ,n4 = crossover (crossover p2,p1,n1,n2),(crossover p2,p1,n1,n2,0 ),n4 by Th15
.= crossover (crossover p2,p1,n1,n2),(crossover p1,p2,n1,n2),n4 by Th15 ;
hence crossover p1,p2,n1,n2,0 ,n4 = crossover p2,p1,n1,n2,n4 ; :: thesis: crossover p1,p2,n1,n2,n3,0 = crossover p2,p1,n1,n2,n3
crossover p1,p2,n1,n2,n3,0 = crossover (crossover p1,p2,n1,n2,n3),(crossover p2,p1,n1,n2,n3),0 ;
hence crossover p1,p2,n1,n2,n3,0 = crossover p2,p1,n1,n2,n3 by Th4; :: thesis: verum