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,n2,n3 = crossover p1,p2,n2,n1,n3 & crossover p1,p2,n1,n2,n3 = crossover p1,p2,n1,n3,n2 )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n2,n3 = crossover p1,p2,n2,n1,n3 & crossover p1,p2,n1,n2,n3 = crossover p1,p2,n1,n3,n2 )
let p1, p2 be Individual of S; :: thesis: ( crossover p1,p2,n1,n2,n3 = crossover p1,p2,n2,n1,n3 & crossover p1,p2,n1,n2,n3 = crossover p1,p2,n1,n3,n2 )
set q1 = crossover p1,p2,n1;
set q2 = crossover p2,p1,n1;
crossover p1,p2,n1,n2,n3 = crossover (crossover p1,p2,n2,n1),(crossover p2,p1,n1,n2),n3 by Th13
.= crossover (crossover p1,p2,n2,n1),(crossover p2,p1,n2,n1),n3 by Th13 ;
hence crossover p1,p2,n1,n2,n3 = crossover p1,p2,n2,n1,n3 ; :: thesis: crossover p1,p2,n1,n2,n3 = crossover p1,p2,n1,n3,n2
crossover p1,p2,n1,n2,n3 = crossover (crossover p1,p2,n1),(crossover p2,p1,n1),n2,n3
.= crossover (crossover p1,p2,n1),(crossover p2,p1,n1),n3,n2 by Th13
.= crossover (crossover p1,p2,n1,n3),(crossover p2,p1,n1,n3),n2 ;
hence crossover p1,p2,n1,n2,n3 = crossover p1,p2,n1,n3,n2 ; :: thesis: verum