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

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