let n1, n3, n4, n5, n2 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n1,n3,n4,n5 = crossover p1,p2,n3,n4,n5 & crossover p1,p2,n1,n2,n1,n4,n5 = crossover p1,p2,n2,n4,n5 & crossover p1,p2,n1,n2,n3,n1,n5 = crossover p1,p2,n2,n3,n5 & crossover p1,p2,n1,n2,n3,n4,n1 = crossover p1,p2,n2,n3,n4 )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n1,n3,n4,n5 = crossover p1,p2,n3,n4,n5 & crossover p1,p2,n1,n2,n1,n4,n5 = crossover p1,p2,n2,n4,n5 & crossover p1,p2,n1,n2,n3,n1,n5 = crossover p1,p2,n2,n3,n5 & crossover p1,p2,n1,n2,n3,n4,n1 = crossover p1,p2,n2,n3,n4 )
let p1, p2 be Individual of S; :: thesis: ( crossover p1,p2,n1,n1,n3,n4,n5 = crossover p1,p2,n3,n4,n5 & crossover p1,p2,n1,n2,n1,n4,n5 = crossover p1,p2,n2,n4,n5 & crossover p1,p2,n1,n2,n3,n1,n5 = crossover p1,p2,n2,n3,n5 & crossover p1,p2,n1,n2,n3,n4,n1 = crossover p1,p2,n2,n3,n4 )
set q1 = crossover p1,p2,n2,n3,n4;
set q2 = crossover p2,p1,n2,n3,n4;
crossover p1,p2,n1,n1,n3,n4,n5 = crossover (crossover p1,p2,n3,n4),(crossover p2,p1,n1,n1,n3,n4),n5 by Th38
.= crossover (crossover p1,p2,n3,n4),(crossover p2,p1,n3,n4),n5 by Th38 ;
hence crossover p1,p2,n1,n1,n3,n4,n5 = crossover p1,p2,n3,n4,n5 ; :: thesis: ( crossover p1,p2,n1,n2,n1,n4,n5 = crossover p1,p2,n2,n4,n5 & crossover p1,p2,n1,n2,n3,n1,n5 = crossover p1,p2,n2,n3,n5 & crossover p1,p2,n1,n2,n3,n4,n1 = crossover p1,p2,n2,n3,n4 )
crossover p1,p2,n1,n2,n1,n4,n5 = crossover (crossover p1,p2,n2,n4),(crossover p2,p1,n1,n2,n1,n4),n5 by Th38
.= crossover (crossover p1,p2,n2,n4),(crossover p2,p1,n2,n4),n5 by Th38 ;
hence crossover p1,p2,n1,n2,n1,n4,n5 = crossover p1,p2,n2,n4,n5 ; :: thesis: ( crossover p1,p2,n1,n2,n3,n1,n5 = crossover p1,p2,n2,n3,n5 & crossover p1,p2,n1,n2,n3,n4,n1 = crossover p1,p2,n2,n3,n4 )
crossover p1,p2,n1,n2,n3,n1,n5 = crossover (crossover p1,p2,n2,n3),(crossover p2,p1,n1,n2,n3,n1),n5 by Th38
.= crossover (crossover p1,p2,n2,n3),(crossover p2,p1,n2,n3),n5 by Th38 ;
hence crossover p1,p2,n1,n2,n3,n1,n5 = crossover p1,p2,n2,n3,n5 ; :: thesis: crossover p1,p2,n1,n2,n3,n4,n1 = crossover p1,p2,n2,n3,n4
crossover p1,p2,n1,n2,n3,n4,n1 = crossover (crossover p1,p2,n2,n3,n4,n1),(crossover p2,p1,n1,n2,n3,n4),n1 by Th37
.= crossover (crossover p1,p2,n2,n3,n4,n1),(crossover p2,p1,n2,n3,n4,n1),n1 by Th37
.= crossover (crossover p1,p2,n2,n3,n4),(crossover p2,p1,n2,n3,n4),n1,n1 ;
hence crossover p1,p2,n1,n2,n3,n4,n1 = crossover p1,p2,n2,n3,n4 by Th12; :: thesis: verum