let n1, n2, n3, n4 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,0,0,0) = crossover (p2,p1,n1) & crossover (p1,p2,0,n2,0,0) = crossover (p2,p1,n2) & crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) & crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,0,0,0) = crossover (p2,p1,n1) & crossover (p1,p2,0,n2,0,0) = crossover (p2,p1,n2) & crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) & crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) )
let p1, p2 be Individual of S; :: thesis: ( crossover (p1,p2,n1,0,0,0) = crossover (p2,p1,n1) & crossover (p1,p2,0,n2,0,0) = crossover (p2,p1,n2) & crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) & crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) )
crossover (p1,p2,n1,0,0,0) = crossover (p1,p2,n1,0) by Th30;
hence crossover (p1,p2,n1,0,0,0) = crossover (p2,p1,n1) by Th8; :: thesis: ( crossover (p1,p2,0,n2,0,0) = crossover (p2,p1,n2) & crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) & crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) )
crossover (p1,p2,0,n2,0,0) = crossover (p1,p2,0,n2) by Th30;
hence crossover (p1,p2,0,n2,0,0) = crossover (p2,p1,n2) by Th7; :: thesis: ( crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) & crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) )
crossover (p1,p2,0,0,n3,0) = crossover (p1,p2,0,n3) by Th30;
hence crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) by Th7; :: thesis: crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4)
crossover (p1,p2,0,0,0,n4) = crossover (p1,p2,0,n4) by Th30;
hence crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) by Th7; :: thesis: verum