let n1, n2, n3 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n3) = p1 & crossover (p1,p2,n1,n2,n1,n2) = p1 & crossover (p1,p2,n1,n2,n2,n1) = p1 )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n3) = p1 & crossover (p1,p2,n1,n2,n1,n2) = p1 & crossover (p1,p2,n1,n2,n2,n1) = p1 )
let p1, p2 be Individual of S; :: thesis: ( crossover (p1,p2,n1,n1,n3,n3) = p1 & crossover (p1,p2,n1,n2,n1,n2) = p1 & crossover (p1,p2,n1,n2,n2,n1) = p1 )
crossover (p1,p2,n1,n1,n3,n3) = crossover (p1,p2,n3,n3) by Th38;
hence crossover (p1,p2,n1,n1,n3,n3) = p1 by Th12; :: thesis: ( crossover (p1,p2,n1,n2,n1,n2) = p1 & crossover (p1,p2,n1,n2,n2,n1) = p1 )
crossover (p1,p2,n1,n2,n1,n2) = crossover (p1,p2,n2,n2) by Th38;
hence crossover (p1,p2,n1,n2,n1,n2) = p1 by Th12; :: thesis: crossover (p1,p2,n1,n2,n2,n1) = p1
crossover (p1,p2,n1,n2,n2,n1) = crossover (p1,p2,n2,n2) by Th38;
hence crossover (p1,p2,n1,n2,n2,n1) = p1 by Th12; :: thesis: verum