reserve D for non empty set;
reserve f1,f2 for FinSequence of D;
reserve i,n,n1,n2,n3,n4,n5,n6 for Element of NAT;
reserve S for Gene-Set;
reserve p1,p2 for Individual of S;

theorem
  (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))
proof
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;
    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 thesis by A4,Th5;
  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;
    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 thesis by A8,Th5;
  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;
    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 thesis by A14,A16,A15,A17,Th36;
  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;
    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 thesis by A21,Th5;
  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;
    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 thesis by A24,Th5;
  end;
  hence thesis by A9,A1,A5,A18;
end;
