let n1, n2, n3, n4, n5 be Element of NAT ; :: thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1) ) )
let S be Gene-Set; :: thesis: for p1, p2 being Individual of S holds
( ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1) ) )
let p1, p2 be Individual of S; :: thesis: ( ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1) ) )
A1: ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3) )

proof

assume that
A2: ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 ) and
A3: n5 >= len p1 ; :: thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3)
n5 >= len S by A3, Def1;
then A4: n5 >= len (crossover (p1,p2,n3)) by Def1;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n3)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A2, Th35;
hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3) by A4, Th5; :: thesis: verum

end;

A5: ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2) )

proof

assume that
A6: ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 ) and
A7: n5 >= len p1 ; :: thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2)
n5 >= len S by A7, Def1;
then A8: n5 >= len (crossover (p1,p2,n2)) by Def1;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A6, Th35;
hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2) by A8, Th5; :: thesis: verum

end;

A9: ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5) )

proof

assume that
A10: n1 >= len p1 and
A11: n2 >= len p1 and
A12: n3 >= len p1 and
A13: n4 >= len p1 ; :: thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5)
n1 >= len S by A10, Def1;
then A14: n1 >= len p2 by Def1;
n3 >= len S by A12, Def1;
then A15: n3 >= len p2 by Def1;
n2 >= len S by A11, Def1;
then A16: n2 >= len p2 by Def1;
n4 >= len S by A13, Def1;
then A17: n4 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,(crossover (p2,p1,n1,n2,n3,n4)),n5) by A10, A11, A12, A13, Th36;
hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5) by A14, A16, A15, A17, Th36; :: thesis: verum

end;

A18: ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1) )

proof

assume that
A19: ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 ) and
A20: n5 >= len p1 ; :: thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1)
n5 >= len S by A20, Def1;
then A21: n5 >= len (crossover (p1,p2,n1)) by Def1;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A19, Th35;
hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1) by A21, Th5; :: thesis: verum

end;

( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4) )

proof

assume that
A22: ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 ) and
A23: n5 >= len p1 ; :: thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4)
n5 >= len S by A23, Def1;
then A24: n5 >= len (crossover (p1,p2,n4)) by Def1;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A22, Th35;
hence crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4) by A24, Th5; :: thesis: verum

end;

hence ( ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5) ) & ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1) ) ) by A9, A1, A5, A18; :: thesis: verum