let n1, n2, n3, n4, n5, n6 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,n6 = crossover p1,p2,n2,n1,n3,n4,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n3,n2,n1,n4,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n4,n2,n3,n1,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n5,n2,n3,n4,n1,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n6,n2,n3,n4,n5,n1 )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n2,n1,n3,n4,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n3,n2,n1,n4,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n4,n2,n3,n1,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n5,n2,n3,n4,n1,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n6,n2,n3,n4,n5,n1 )
let p1, p2 be Individual of S; :: thesis: ( crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n2,n1,n3,n4,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n3,n2,n1,n4,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n4,n2,n3,n1,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n5,n2,n3,n4,n1,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n6,n2,n3,n4,n5,n1 )
set q1 = crossover p1,p2,n5,n2,n3,n4;
set q2 = crossover p2,p1,n5,n2,n3,n4;
A1: crossover p1,p2,n5,n2,n3,n4,n6 = crossover (crossover p1,p2,n5,n2,n3,n4),(crossover p2,p1,n5,n2,n3,n4),n6 ;
crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover (crossover p1,p2,n2,n1,n3,n4,n5),(crossover p2,p1,n1,n2,n3,n4,n5),n6 by Th51
.= crossover (crossover p1,p2,n2,n1,n3,n4,n5),(crossover p2,p1,n2,n1,n3,n4,n5),n6 by Th51 ;
hence crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n2,n1,n3,n4,n5,n6 ; :: thesis: ( crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n3,n2,n1,n4,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n4,n2,n3,n1,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n5,n2,n3,n4,n1,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n6,n2,n3,n4,n5,n1 )
crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover (crossover p1,p2,n3,n2,n1,n4,n5),(crossover p2,p1,n1,n2,n3,n4,n5),n6 by Th51
.= crossover (crossover p1,p2,n3,n2,n1,n4,n5),(crossover p2,p1,n3,n2,n1,n4,n5),n6 by Th51 ;
hence crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n3,n2,n1,n4,n5,n6 ; :: thesis: ( crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n4,n2,n3,n1,n5,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n5,n2,n3,n4,n1,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n6,n2,n3,n4,n5,n1 )
crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover (crossover p1,p2,n4,n2,n3,n1,n5),(crossover p2,p1,n1,n2,n3,n4,n5),n6 by Th51
.= crossover (crossover p1,p2,n4,n2,n3,n1,n5),(crossover p2,p1,n4,n2,n3,n1,n5),n6 by Th51 ;
hence crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n4,n2,n3,n1,n5,n6 ; :: thesis: ( crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n5,n2,n3,n4,n1,n6 & crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n6,n2,n3,n4,n5,n1 )
crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover (crossover p1,p2,n5,n2,n3,n4,n1),(crossover p2,p1,n1,n2,n3,n4,n5),n6 by Th51
.= crossover (crossover p1,p2,n5,n2,n3,n4,n1),(crossover p2,p1,n5,n2,n3,n4,n1),n6 by Th51 ;
hence crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n5,n2,n3,n4,n1,n6 ; :: thesis: crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n6,n2,n3,n4,n5,n1
crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover (crossover p1,p2,n5,n2,n3,n4,n1),(crossover p2,p1,n1,n2,n3,n4,n5),n6 by Th51
.= crossover (crossover p1,p2,n5,n2,n3,n4,n1),(crossover p2,p1,n5,n2,n3,n4,n1),n6 by Th51
.= crossover (crossover p1,p2,n5,n2,n3,n4),(crossover p2,p1,n5,n2,n3,n4),n1,n6
.= crossover (crossover p1,p2,n5,n2,n3,n4),(crossover p2,p1,n5,n2,n3,n4),n6,n1 by Th13
.= crossover (crossover p1,p2,n6,n2,n3,n4,n5),(crossover p2,p1,n5,n2,n3,n4,n6),n1 by A1, Th51
.= crossover (crossover p1,p2,n6,n2,n3,n4,n5),(crossover p2,p1,n6,n2,n3,n4,n5),n1 by Th51 ;
hence crossover p1,p2,n1,n2,n3,n4,n5,n6 = crossover p1,p2,n6,n2,n3,n4,n5,n1 ; :: thesis: verum