let n1, n2, n3 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S st n2 >= len p1 & n3 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1)
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S st n2 >= len p1 & n3 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1)
let p1, p2 be Individual of S; :: thesis: ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1) )
assume that
A1: n2 >= len p1 and
A2: n3 >= len p1 ; :: thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1)
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3) by A1, Th19;
hence crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1) by A2, Th10; :: thesis: verum