let n1, n2, n3, n4, n5 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n1,n3,n4,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n3,n2,n1,n4,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n4,n2,n3,n1,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n5,n2,n3,n4,n1 )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n1,n3,n4,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n3,n2,n1,n4,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n4,n2,n3,n1,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n5,n2,n3,n4,n1 )
let p1, p2 be Individual of S; :: thesis: ( crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n1,n3,n4,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n3,n2,n1,n4,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n4,n2,n3,n1,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n5,n2,n3,n4,n1 )
A1: crossover p1,p2,n1,n2,n3,n4,n5 = crossover (crossover p1,p2,n3,n2,n1,n4),(crossover p2,p1,n1,n2,n3,n4),n5 by Th37
.= crossover (crossover p1,p2,n3,n2,n1,n4),(crossover p2,p1,n3,n2,n1,n4),n5 by Th37 ;
A2: crossover p1,p2,n1,n2,n3,n4,n5 = crossover (crossover p1,p2,n4,n2,n3,n1),(crossover p2,p1,n1,n2,n3,n4),n5 by Th37
.= crossover (crossover p1,p2,n4,n2,n3,n1),(crossover p2,p1,n4,n2,n3,n1),n5 by Th37 ;
A3: crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n5,n2,n3,n4,n1

proof

set q2 = crossover p2,p1,n2,n3,n4;
set q1 = crossover p1,p2,n2,n3,n4;
A4: crossover p1,p2,n2,n3,n4,n5 = crossover (crossover p1,p2,n2,n3,n4),(crossover p2,p1,n2,n3,n4),n5 ;
crossover p1,p2,n1,n2,n3,n4,n5 = crossover (crossover p1,p2,n2,n3,n4,n1),(crossover p2,p1,n1,n2,n3,n4),n5 by Th37
.= crossover (crossover p1,p2,n2,n3,n4,n1),(crossover p2,p1,n2,n3,n4,n1),n5 by Th37
.= crossover (crossover p1,p2,n2,n3,n4),(crossover p2,p1,n2,n3,n4),n1,n5
.= crossover (crossover p1,p2,n2,n3,n4),(crossover p2,p1,n2,n3,n4),n5,n1 by Th13
.= crossover (crossover p1,p2,n5,n2,n3,n4),(crossover p2,p1,n2,n3,n4,n5),n1 by A4, Th37
.= crossover (crossover p1,p2,n5,n2,n3,n4),(crossover p2,p1,n5,n2,n3,n4),n1 by Th37 ;
hence crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n5,n2,n3,n4,n1 ; :: thesis: verum

end;

crossover p1,p2,n1,n2,n3,n4,n5 = crossover (crossover p1,p2,n2,n1,n3,n4),(crossover p2,p1,n1,n2,n3,n4),n5 by Th37
.= crossover (crossover p1,p2,n2,n1,n3,n4),(crossover p2,p1,n2,n1,n3,n4),n5 by Th37 ;
hence ( crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n2,n1,n3,n4,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n3,n2,n1,n4,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n4,n2,n3,n1,n5 & crossover p1,p2,n1,n2,n3,n4,n5 = crossover p1,p2,n5,n2,n3,n4,n1 ) by A1, A2, A3; :: thesis: verum