let n be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S st n >= len p1 holds
crossover (p1,p2,n) = p1
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S st n >= len p1 holds
crossover (p1,p2,n) = p1
let p1, p2 be Individual of S; :: thesis: ( n >= len p1 implies crossover (p1,p2,n) = p1 )
assume A1: n >= len p1 ; :: thesis: crossover (p1,p2,n) = p1
then n >= len S by Def1;
then A2: n >= len p2 by Def1;
crossover (p1,p2,n) = p1 ^ (p2 /^ n) by A1, FINSEQ_1:58
.= p1 ^ {} by A2, FINSEQ_5:32
.= p1 by FINSEQ_1:34 ;
hence crossover (p1,p2,n) = p1 ; :: thesis: verum