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 implies crossover(p1,p2,n1,n2,n3
,n4,n5)=crossover(p1,p2,n4,n5)) & (n1>=len p1 & n2>=len p1 & n4>=len p1 implies
crossover(p1,p2,n1,n2,n3,n4,n5)=crossover(p1,p2,n3,n5)) & (n1>=len p1 & n2>=len
p1 & n5>=len p1 implies crossover(p1,p2,n1,n2,n3,n4,n5)=crossover(p1,p2,n3,n4))
& (n1>=len p1 & n3>=len p1 & n4>=len p1 implies crossover(p1,p2,n1,n2,n3,n4,n5)
  =crossover(p1,p2,n2,n5)) & (n1>=len p1 & n3>=len p1 & n5>=len p1 implies
crossover(p1,p2,n1,n2,n3,n4,n5)=crossover(p1,p2,n2,n4)) & (n1>=len p1 & n4>=len
p1 & n5>=len p1 implies crossover(p1,p2,n1,n2,n3,n4,n5)=crossover(p1,p2,n2,n3))
& (n2>=len p1 & n3>=len p1 & n4>=len p1 implies crossover(p1,p2,n1,n2,n3,n4,n5)
  =crossover(p1,p2,n1,n5)) & (n2>=len p1 & n3>=len p1 & n5>=len p1 implies
crossover(p1,p2,n1,n2,n3,n4,n5)=crossover(p1,p2,n1,n4)) & (n2>=len p1 & n4>=len
p1 & n5>=len p1 implies crossover(p1,p2,n1,n2,n3,n4,n5)=crossover(p1,p2,n1,n3))
& (n3>=len p1 & n4>=len p1 & n5>=len p1 implies crossover(p1,p2,n1,n2,n3,n4,n5)
  =crossover(p1,p2,n1,n2))
proof
A1: n1>=len p1 & n3>=len p1 & n5>=len p1 implies crossover(p1,p2,n1,n2,n3,
  n4,n5)=crossover(p1,p2,n2,n4)
  proof
    assume that
A2: n1>=len p1 & n3>=len p1 and
A3: n5>=len p1;
    n5>=len S by A3,Def1;
    then
A4: n5>=len crossover(p1,p2,n2,n4) by Def1;
    crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(crossover(p1,p2,n2,n4),
    crossover(p2,p1,n1,n2,n3,n4),n5) by A2,Th34;
    hence thesis by A4,Th5;
  end;
A5: n1>=len p1 & n4>=len p1 & n5>=len p1 implies crossover(p1,p2,n1,n2,n3,
  n4,n5)=crossover(p1,p2,n2,n3)
  proof
    assume that
A6: n1>=len p1 & n4>=len p1 and
A7: n5>=len p1;
    n5>=len S by A7,Def1;
    then
A8: n5>=len crossover(p1,p2,n2,n3) by Def1;
    crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(crossover(p1,p2,n2,n3),
    crossover(p2,p1,n1,n2,n3,n4),n5) by A6,Th34;
    hence thesis by A8,Th5;
  end;
A9: n1>=len p1 & n2>=len p1 & n4>=len p1 implies crossover(p1,p2,n1,n2,n3,
  n4,n5)=crossover(p1,p2,n3,n5)
  proof
    assume that
A10: n1>=len p1 and
A11: n2>=len p1 and
A12: n4>=len p1;
    n1>=len S by A10,Def1;
    then
A13: n1>=len p2 by Def1;
    n4>=len S by A12,Def1;
    then
A14: n4>=len p2 by Def1;
    n2>=len S by A11,Def1;
    then
A15: n2>=len p2 by Def1;
    crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(crossover(p1,p2,n3),
    crossover(p2,p1,n1,n2,n3,n4),n5) by A10,A11,A12,Th35
      .=crossover(crossover(p1,p2,n3),crossover(p2,p1,n3),n5) by A13,A15,A14
,Th35;
    hence thesis;
  end;
A16: n2>=len p1 & n4>=len p1 & n5>=len p1 implies crossover(p1,p2,n1,n2,n3,
  n4,n5)=crossover(p1,p2,n1,n3)
  proof
    assume that
A17: n2>=len p1 & n4>=len p1 and
A18: n5>=len p1;
    n5>=len S by A18,Def1;
    then
A19: n5>=len crossover(p1,p2,n1,n3) by Def1;
    crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(crossover(p1,p2,n1,n3),
    crossover(p2,p1,n1,n2,n3,n4),n5) by A17,Th34;
    hence thesis by A19,Th5;
  end;
A20: n2>=len p1 & n3>=len p1 & n4>=len p1 implies crossover(p1,p2,n1,n2,n3,
  n4,n5)=crossover(p1,p2,n1,n5)
  proof
    assume that
A21: n2>=len p1 and
A22: n3>=len p1 and
A23: n4>=len p1;
    n2>=len S by A21,Def1;
    then
A24: n2>=len p2 by Def1;
    n4>=len S by A23,Def1;
    then
A25: n4>=len p2 by Def1;
    n3>=len S by A22,Def1;
    then
A26: n3>=len p2 by Def1;
    crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(crossover(p1,p2,n1),
    crossover(p2,p1,n1,n2,n3,n4),n5) by A21,A22,A23,Th35
      .=crossover(crossover(p1,p2,n1),crossover(p2,p1,n1),n5) by A24,A26,A25
,Th35;
    hence thesis;
  end;
A27: n1>=len p1 & n2>=len p1 & n3>=len p1 implies crossover(p1,p2,n1,n2,n3,n4
  ,n5)=crossover(p1,p2,n4,n5)
  proof
    assume that
A28: n1>=len p1 and
A29: n2>=len p1 and
A30: n3>=len p1;
    n1>=len S by A28,Def1;
    then
A31: n1>=len p2 by Def1;
    n3>=len S by A30,Def1;
    then
A32: n3>=len p2 by Def1;
    n2>=len S by A29,Def1;
    then
A33: n2>=len p2 by Def1;
    crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(crossover(p1,p2,n4),
    crossover(p2,p1,n1,n2,n3,n4),n5) by A28,A29,A30,Th35
      .=crossover(crossover(p1,p2,n4),crossover(p2,p1,n4),n5) by A31,A33,A32
,Th35;
    hence thesis;
  end;
A34: n2>=len p1 & n3>=len p1 & n5>=len p1 implies crossover(p1,p2,n1,n2,n3,
  n4,n5)=crossover(p1,p2,n1,n4)
  proof
    assume that
A35: n2>=len p1 & n3>=len p1 and
A36: n5>=len p1;
    n5>=len S by A36,Def1;
    then
A37: n5>=len crossover(p1,p2,n1,n4) by Def1;
    crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(crossover(p1,p2,n1,n4),
    crossover(p2,p1,n1,n2,n3,n4),n5) by A35,Th34;
    hence thesis by A37,Th5;
  end;
A38: n1>=len p1 & n3>=len p1 & n4>=len p1 implies crossover(p1,p2,n1,n2,n3,
  n4,n5)=crossover(p1,p2,n2,n5)
  proof
    assume that
A39: n1>=len p1 and
A40: n3>=len p1 and
A41: n4>=len p1;
    n1>=len S by A39,Def1;
    then
A42: n1>=len p2 by Def1;
    n4>=len S by A41,Def1;
    then
A43: n4>=len p2 by Def1;
    n3>=len S by A40,Def1;
    then
A44: n3>=len p2 by Def1;
    crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(crossover(p1,p2,n2),
    crossover(p2,p1,n1,n2,n3,n4),n5) by A39,A40,A41,Th35
      .=crossover(crossover(p1,p2,n2),crossover(p2,p1,n2),n5) by A42,A44,A43
,Th35;
    hence thesis;
  end;
A45: n3>=len p1 & n4>=len p1 & n5>=len p1 implies crossover(p1,p2,n1,n2,n3,
  n4,n5)=crossover(p1,p2,n1,n2)
  proof
    assume that
A46: n3>=len p1 & n4>=len p1 and
A47: n5>=len p1;
    n5>=len S by A47,Def1;
    then
A48: n5>=len crossover(p1,p2,n1,n2) by Def1;
    crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(crossover(p1,p2,n1,n2),
    crossover(p2,p1,n1,n2,n3,n4),n5) by A46,Th34;
    hence thesis by A48,Th5;
  end;
  n1>=len p1 & n2>=len p1 & n5>=len p1 implies crossover(p1,p2,n1,n2,n3,
  n4,n5)=crossover(p1,p2,n3,n4)
  proof
    assume that
A49: n1>=len p1 & n2>=len p1 and
A50: n5>=len p1;
    n5>=len S by A50,Def1;
    then
A51: n5>=len crossover(p1,p2,n3,n4) by Def1;
    crossover(p1,p2,n1,n2,n3,n4,n5) =crossover(crossover(p1,p2,n3,n4),
    crossover(p2,p1,n1,n2,n3,n4),n5) by A49,Th34;
    hence thesis by A51,Th5;
  end;
  hence thesis by A27,A9,A38,A1,A5,A20,A34,A16,A45;
end;
