let n1, n2 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 holds
crossover (p1,p2,n1,n2) = p1
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 holds
crossover (p1,p2,n1,n2) = p1
let p1, p2 be Individual of S; :: thesis: ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2) = p1 )
assume that
A1: n1 >= len p1 and
A2: n2 >= len p1 ; :: thesis: crossover (p1,p2,n1,n2) = p1
crossover (p1,p2,n1,n2) = crossover (p1,p2,n2) by A1, Th9;
hence crossover (p1,p2,n1,n2) = p1 by A2, Th5; :: thesis: verum