let n be Element of NAT ; 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; for p1, p2 being Individual of S st n >= len p1 holds
crossover (p1,p2,n) = p1
let p1, p2 be Individual of S; ( n >= len p1 implies crossover (p1,p2,n) = p1 )
assume A1:
n >= len p1
; 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
; verum