let n1, n3, n2 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n1,n3 = crossover p1,p2,n3 & crossover p1,p2,n1,n2,n1 = crossover p1,p2,n2 & crossover p1,p2,n1,n2,n2 = crossover p1,p2,n1 )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n1,n3 = crossover p1,p2,n3 & crossover p1,p2,n1,n2,n1 = crossover p1,p2,n2 & crossover p1,p2,n1,n2,n2 = crossover p1,p2,n1 )
let p1, p2 be Individual of S; :: thesis: ( crossover p1,p2,n1,n1,n3 = crossover p1,p2,n3 & crossover p1,p2,n1,n2,n1 = crossover p1,p2,n2 & crossover p1,p2,n1,n2,n2 = crossover p1,p2,n1 )
crossover p1,p2,n1,n1,n3 = crossover p1,(crossover p2,p1,n1,n1),n3 by Th12;
hence crossover p1,p2,n1,n1,n3 = crossover p1,p2,n3 by Th12; :: thesis: ( crossover p1,p2,n1,n2,n1 = crossover p1,p2,n2 & crossover p1,p2,n1,n2,n2 = crossover p1,p2,n1 )
crossover p1,p2,n1,n2,n1 = crossover p1,p2,n1,n1,n2 by Th25
.= crossover p1,(crossover p2,p1,n1,n1),n2 by Th12 ;
hence crossover p1,p2,n1,n2,n1 = crossover p1,p2,n2 by Th12; :: thesis: crossover p1,p2,n1,n2,n2 = crossover p1,p2,n1
thus crossover p1,p2,n1,n2,n2 = crossover p1,p2,n2,n2,n1 by Th26
.= crossover p1,(crossover p2,p1,n2,n2),n1 by Th12
.= crossover p1,p2,n1 by Th12 ; :: thesis: verum