let n1, n3, n4, 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 = crossover p1,p2,n3,n4 & crossover p1,p2,n1,n2,n1,n4 = crossover p1,p2,n2,n4 & crossover p1,p2,n1,n2,n3,n1 = crossover p1,p2,n2,n3 & crossover p1,p2,n1,n2,n2,n4 = crossover p1,p2,n1,n4 & crossover p1,p2,n1,n2,n3,n2 = crossover p1,p2,n1,n3 & crossover p1,p2,n1,n2,n3,n3 = crossover p1,p2,n1,n2 )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n1,n3,n4 = crossover p1,p2,n3,n4 & crossover p1,p2,n1,n2,n1,n4 = crossover p1,p2,n2,n4 & crossover p1,p2,n1,n2,n3,n1 = crossover p1,p2,n2,n3 & crossover p1,p2,n1,n2,n2,n4 = crossover p1,p2,n1,n4 & crossover p1,p2,n1,n2,n3,n2 = crossover p1,p2,n1,n3 & crossover p1,p2,n1,n2,n3,n3 = crossover p1,p2,n1,n2 )
let p1, p2 be Individual of S; :: thesis: ( crossover p1,p2,n1,n1,n3,n4 = crossover p1,p2,n3,n4 & crossover p1,p2,n1,n2,n1,n4 = crossover p1,p2,n2,n4 & crossover p1,p2,n1,n2,n3,n1 = crossover p1,p2,n2,n3 & crossover p1,p2,n1,n2,n2,n4 = crossover p1,p2,n1,n4 & crossover p1,p2,n1,n2,n3,n2 = crossover p1,p2,n1,n3 & crossover p1,p2,n1,n2,n3,n3 = crossover p1,p2,n1,n2 )
crossover p1,p2,n1,n1,n3,n4 = crossover (crossover p1,p2,n3),(crossover p2,p1,n1,n1,n3),n4 by Th27
.= crossover (crossover p1,p2,n3),(crossover p2,p1,n3),n4 by Th27 ;
hence crossover p1,p2,n1,n1,n3,n4 = crossover p1,p2,n3,n4 ; :: thesis: ( crossover p1,p2,n1,n2,n1,n4 = crossover p1,p2,n2,n4 & crossover p1,p2,n1,n2,n3,n1 = crossover p1,p2,n2,n3 & crossover p1,p2,n1,n2,n2,n4 = crossover p1,p2,n1,n4 & crossover p1,p2,n1,n2,n3,n2 = crossover p1,p2,n1,n3 & crossover p1,p2,n1,n2,n3,n3 = crossover p1,p2,n1,n2 )
crossover p1,p2,n1,n2,n1,n4 = crossover (crossover p1,p2,n2),(crossover p2,p1,n1,n2,n1),n4 by Th27
.= crossover (crossover p1,p2,n2),(crossover p2,p1,n2),n4 by Th27 ;
hence crossover p1,p2,n1,n2,n1,n4 = crossover p1,p2,n2,n4 ; :: thesis: ( crossover p1,p2,n1,n2,n3,n1 = crossover p1,p2,n2,n3 & crossover p1,p2,n1,n2,n2,n4 = crossover p1,p2,n1,n4 & crossover p1,p2,n1,n2,n3,n2 = crossover p1,p2,n1,n3 & crossover p1,p2,n1,n2,n3,n3 = crossover p1,p2,n1,n2 )
crossover p1,p2,n1,n2,n3,n1 = crossover p1,p2,n1,n1,n2,n3 by Th37
.= crossover (crossover p1,p2,n2),(crossover p2,p1,n1,n1,n2),n3 by Th27
.= crossover (crossover p1,p2,n2),(crossover p2,p1,n2),n3 by Th27 ;
hence crossover p1,p2,n1,n2,n3,n1 = crossover p1,p2,n2,n3 ; :: thesis: ( crossover p1,p2,n1,n2,n2,n4 = crossover p1,p2,n1,n4 & crossover p1,p2,n1,n2,n3,n2 = crossover p1,p2,n1,n3 & crossover p1,p2,n1,n2,n3,n3 = crossover p1,p2,n1,n2 )
crossover p1,p2,n1,n2,n2,n4 = crossover (crossover p1,p2,n1),(crossover p2,p1,n1,n2,n2),n4 by Th27
.= crossover (crossover p1,p2,n1),(crossover p2,p1,n1),n4 by Th27 ;
hence crossover p1,p2,n1,n2,n2,n4 = crossover p1,p2,n1,n4 ; :: thesis: ( crossover p1,p2,n1,n2,n3,n2 = crossover p1,p2,n1,n3 & crossover p1,p2,n1,n2,n3,n3 = crossover p1,p2,n1,n2 )
crossover p1,p2,n1,n2,n3,n2 = crossover p1,p2,n1,n2,n2,n3 by Th37
.= crossover (crossover p1,p2,n1),(crossover p2,p1,n1,n2,n2),n3 by Th27
.= crossover (crossover p1,p2,n1),(crossover p2,p1,n1),n3 by Th27 ;
hence crossover p1,p2,n1,n2,n3,n2 = crossover p1,p2,n1,n3 ; :: thesis: crossover p1,p2,n1,n2,n3,n3 = crossover p1,p2,n1,n2
crossover p1,p2,n1,n2,n3,n3 = crossover p1,p2,n1,n3,n3,n2 by Th37
.= crossover (crossover p1,p2,n1),(crossover p2,p1,n1,n3,n3),n2 by Th27
.= crossover (crossover p1,p2,n1),(crossover p2,p1,n1),n2 by Th27 ;
hence crossover p1,p2,n1,n2,n3,n3 = crossover p1,p2,n1,n2 ; :: thesis: verum