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 Th37:
  crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n1,n2,n4,n3) &
crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n1,n3,n2,n4) & crossover(p1,p2,
  n1,n2,n3,n4) = crossover(p1,p2,n1,n3,n4,n2) & crossover(p1,p2,n1,n2,n3,n4) =
crossover(p1,p2,n1,n4,n3,n2) & crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,
  n2,n1,n3,n4) & crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n2,n1,n4,n3) &
crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n2,n3,n1,n4) & crossover(p1,p2,
  n1,n2,n3,n4) = crossover(p1,p2,n2,n3,n4,n1) & crossover(p1,p2,n1,n2,n3,n4) =
crossover(p1,p2,n2,n4,n1,n3) & crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,
  n2,n4,n3,n1) & crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n3,n2,n1,n4) &
crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n3,n2,n4,n1) & crossover(p1,p2,
  n1,n2,n3,n4) = crossover(p1,p2,n3,n4,n1,n2) & crossover(p1,p2,n1,n2,n3,n4) =
crossover(p1,p2,n3,n4,n2,n1) & crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,
  n4,n2,n3,n1) & crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n4,n3,n2,n1)
proof
A1: crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n1,n2,n4,n3)
  proof
    set q2=crossover(p2,p1,n1,n2);
    set q1=crossover(p1,p2,n1,n2);
    crossover(p1,p2,n1,n2,n3,n4) =crossover(q1,q2,n3,n4)
      .=crossover(q1,q2,n4,n3) by Th13
      .=crossover(crossover(q1,q2,n4),crossover(q2,q1,n4),n3);
    hence thesis;
  end;
A2: crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n1,n3,n2),
  crossover(p2,p1,n1,n2,n3),n4) by Th25
    .=crossover(crossover(p1,p2,n1,n3,n2),crossover(p2,p1,n1,n3,n2),n4) by Th25
;
A3: crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n2,n3,n4,n1)
  proof
    set q2=crossover(p2,p1,n2,n3);
    set q1=crossover(p1,p2,n2,n3);
    crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n2,n1,n3),
    crossover(p2,p1,n1,n2,n3),n4) by Th25
      .=crossover(crossover(p1,p2,n2,n3,n1),crossover(p2,p1,n1,n2,n3),n4) by
Th25
      .=crossover(crossover(p1,p2,n2,n3,n1),crossover(p2,p1,n2,n1,n3),n4) by
Th25
      .=crossover(crossover(p1,p2,n2,n3,n1),crossover(p2,p1,n2,n3,n1),n4) by
Th25
      .=crossover(q1,q2,n1,n4)
      .=crossover(q1,q2,n4,n1) by Th13
      .=crossover(crossover(p1,p2,n2,n3,n4),crossover(p2,p1,n2,n3,n4),n1);
    hence thesis;
  end;
A4: crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n1,n4,n3,n2)
  proof
    set q2=crossover(p2,p1,n1,n3);
    set q1=crossover(p1,p2,n1,n3);
    crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n1,n3,n2),
    crossover(p2,p1,n1,n2,n3),n4) by Th25
      .=crossover(crossover(p1,p2,n1,n3,n2),crossover(p2,p1,n1,n3,n2),n4) by
Th25
      .=crossover(q1,q2,n2,n4)
      .=crossover(q1,q2,n4,n2) by Th13
      .=crossover(crossover(p1,p2,n1,n3,n4),crossover(p2,p1,n1,n3,n4),n2)
      .=crossover(crossover(p1,p2,n1,n4,n3),crossover(p2,p1,n1,n3,n4),n2) by
Th25
      .=crossover(crossover(p1,p2,n1,n4,n3),crossover(p2,p1,n1,n4,n3),n2) by
Th25;
    hence thesis;
  end;
A5: crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n4,n3,n2,n1)
  proof
    set q2=crossover(p2,p1,n3,n2);
    set q1=crossover(p1,p2,n3,n2);
    crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n3,n1,n2),
    crossover(p2,p1,n1,n2,n3),n4) by Th26
      .=crossover(crossover(p1,p2,n3,n1,n2),crossover(p2,p1,n3,n1,n2),n4) by
Th26
      .=crossover(crossover(p1,p2,n3,n2,n1),crossover(p2,p1,n3,n1,n2),n4) by
Th25
      .=crossover(crossover(p1,p2,n3,n2,n1),crossover(p2,p1,n3,n2,n1),n4) by
Th25
      .=crossover(q1,q2,n1,n4)
      .=crossover(q1,q2,n4,n1) by Th13
      .=crossover(crossover(p1,p2,n3,n2,n4),crossover(p2,p1,n3,n2,n4),n1)
      .=crossover(crossover(p1,p2,n4,n3,n2),crossover(p2,p1,n3,n2,n4),n1) by
Th26
      .=crossover(crossover(p1,p2,n4,n3,n2),crossover(p2,p1,n4,n3,n2),n1) by
Th26;
    hence thesis;
  end;
A6: crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n2,n1,n4,n3)
  proof
    set q2=crossover(p2,p1,n2,n1);
    set q1=crossover(p1,p2,n2,n1);
    crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n2,n1,n3),
    crossover(p2,p1,n1,n2,n3),n4) by Th25
      .=crossover(crossover(p1,p2,n2,n1,n3),crossover(p2,p1,n2,n1,n3),n4) by
Th25
      .=crossover(q1,q2,n3,n4)
      .=crossover(q1,q2,n4,n3) by Th13
      .=crossover(crossover(p1,p2,n2,n1,n4),crossover(p2,p1,n2,n1,n4),n3);
    hence thesis;
  end;
A7: crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n4,n2,n3,n1)
  proof
    set q2=crossover(p2,p1,n2,n3);
    set q1=crossover(p1,p2,n2,n3);
    crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n2,n1,n3),
    crossover(p2,p1,n1,n2,n3),n4) by Th25
      .=crossover(crossover(p1,p2,n2,n1,n3),crossover(p2,p1,n2,n1,n3),n4) by
Th25
      .=crossover(crossover(p1,p2,n2,n3,n1),crossover(p2,p1,n2,n1,n3),n4) by
Th25
      .=crossover(crossover(p1,p2,n2,n3,n1),crossover(p2,p1,n2,n3,n1),n4) by
Th25
      .=crossover(q1,q2,n1,n4)
      .=crossover(q1,q2,n4,n1) by Th13
      .=crossover(crossover(p1,p2,n2,n3,n4),crossover(p2,p1,n2,n3,n4),n1)
      .=crossover(crossover(p1,p2,n4,n2,n3),crossover(p2,p1,n2,n3,n4),n1) by
Th26
      .=crossover(crossover(p1,p2,n4,n2,n3),crossover(p2,p1,n4,n2,n3),n1) by
Th26;
    hence thesis;
  end;
A8: crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n1,n3,n4,n2)
  proof
    set q2=crossover(p2,p1,n1,n3);
    set q1=crossover(p1,p2,n1,n3);
    crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n1,n3,n2),
    crossover(p2,p1,n1,n2,n3),n4) by Th25
      .=crossover(crossover(p1,p2,n1,n3,n2),crossover(p2,p1,n1,n3,n2),n4) by
Th25
      .=crossover(q1,q2,n2,n4)
      .=crossover(q1,q2,n4,n2) by Th13
      .=crossover(crossover(q1,q2,n4),crossover(q2,q1,n4),n2);
    hence thesis;
  end;
A9: crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n3,n2,n4,n1)
  proof
    set q2=crossover(p2,p1,n3,n2);
    set q1=crossover(p1,p2,n3,n2);
    crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n3,n1,n2),
    crossover(p2,p1,n1,n2,n3),n4) by Th26
      .=crossover(crossover(p1,p2,n3,n1,n2),crossover(p2,p1,n3,n1,n2),n4) by
Th26
      .=crossover(crossover(p1,p2,n3,n2,n1),crossover(p2,p1,n3,n1,n2),n4) by
Th25
      .=crossover(crossover(p1,p2,n3,n2,n1),crossover(p2,p1,n3,n2,n1),n4) by
Th25
      .=crossover(q1,q2,n1,n4)
      .=crossover(q1,q2,n4,n1) by Th13
      .=crossover(crossover(p1,p2,n3,n2,n4),crossover(p2,p1,n3,n2,n4),n1);
    hence thesis;
  end;
A10: crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n2,n4,n1,n3)
  proof
    set q2=crossover(p2,p1,n2,n1);
    set q1=crossover(p1,p2,n2,n1);
    crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n2,n1,n3),
    crossover(p2,p1,n1,n2,n3),n4) by Th25
      .=crossover(crossover(p1,p2,n2,n1,n3),crossover(p2,p1,n2,n1,n3),n4) by
Th25
      .=crossover(q1,q2,n3,n4)
      .=crossover(q1,q2,n4,n3) by Th13
      .=crossover(crossover(p1,p2,n2,n1,n4),crossover(p2,p1,n2,n1,n4),n3)
      .=crossover(crossover(p1,p2,n2,n4,n1),crossover(p2,p1,n2,n1,n4),n3) by
Th25
      .=crossover(crossover(p1,p2,n2,n4,n1),crossover(p2,p1,n2,n4,n1),n3) by
Th25;
    hence thesis;
  end;
A11: crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n3,n4,n2,n1)
  proof
    set q2=crossover(p2,p1,n3,n2);
    set q1=crossover(p1,p2,n3,n2);
    crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n3,n1,n2),
    crossover(p2,p1,n1,n2,n3),n4) by Th26
      .=crossover(crossover(p1,p2,n3,n1,n2),crossover(p2,p1,n3,n1,n2),n4) by
Th26
      .=crossover(crossover(p1,p2,n3,n2,n1),crossover(p2,p1,n3,n1,n2),n4) by
Th25
      .=crossover(crossover(p1,p2,n3,n2,n1),crossover(p2,p1,n3,n2,n1),n4) by
Th25
      .=crossover(q1,q2,n1,n4)
      .=crossover(q1,q2,n4,n1) by Th13
      .=crossover(crossover(p1,p2,n3,n2,n4),crossover(p2,p1,n3,n2,n4),n1)
      .=crossover(crossover(p1,p2,n3,n4,n2),crossover(p2,p1,n3,n2,n4),n1) by
Th25
      .=crossover(crossover(p1,p2,n3,n4,n2),crossover(p2,p1,n3,n4,n2),n1) by
Th25;
    hence thesis;
  end;
A12: crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n3,n1,n2),
  crossover(p2,p1,n1,n2,n3),n4) by Th26
    .=crossover(crossover(p1,p2,n3,n1,n2),crossover(p2,p1,n3,n1,n2),n4) by Th26
    .=crossover(crossover(p1,p2,n3,n2,n1),crossover(p2,p1,n3,n1,n2),n4) by Th25
    .=crossover(crossover(p1,p2,n3,n2,n1),crossover(p2,p1,n3,n2,n1),n4) by Th25
;
A13: crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n3,n4,n1,n2)
  proof
    set q2=crossover(p2,p1,n3,n1);
    set q1=crossover(p1,p2,n3,n1);
    crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n3,n1,n2),
    crossover(p2,p1,n1,n2,n3),n4) by Th26
      .=crossover(crossover(p1,p2,n3,n1,n2),crossover(p2,p1,n3,n1,n2),n4) by
Th26
      .=crossover(q1,q2,n2,n4)
      .=crossover(q1,q2,n4,n2) by Th13
      .=crossover(crossover(p1,p2,n3,n1,n4),crossover(p2,p1,n3,n1,n4),n2)
      .=crossover(crossover(p1,p2,n3,n4,n1),crossover(p2,p1,n3,n1,n4),n2) by
Th25
      .=crossover(crossover(p1,p2,n3,n4,n1),crossover(p2,p1,n3,n4,n1),n2) by
Th25;
    hence thesis;
  end;
A14: crossover(p1,p2,n1,n2,n3,n4) = crossover(p1,p2,n2,n4,n3,n1)
  proof
    set q2=crossover(p2,p1,n2,n3);
    set q1=crossover(p1,p2,n2,n3);
    crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n2,n1,n3),
    crossover(p2,p1,n1,n2,n3),n4) by Th25
      .=crossover(crossover(p1,p2,n2,n3,n1),crossover(p2,p1,n1,n2,n3),n4) by
Th25
      .=crossover(crossover(p1,p2,n2,n3,n1),crossover(p2,p1,n2,n1,n3),n4) by
Th25
      .=crossover(crossover(p1,p2,n2,n3,n1),crossover(p2,p1,n2,n3,n1),n4) by
Th25
      .=crossover(q1,q2,n1,n4)
      .=crossover(q1,q2,n4,n1) by Th13
      .=crossover(crossover(p1,p2,n2,n3,n4),crossover(p2,p1,n2,n3,n4),n1)
      .=crossover(crossover(p1,p2,n2,n4,n3),crossover(p2,p1,n2,n3,n4),n1) by
Th25
      .=crossover(crossover(p1,p2,n2,n4,n3),crossover(p2,p1,n2,n4,n3),n1) by
Th25;
    hence thesis;
  end;
A15: crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n2,n1,n3),
  crossover(p2,p1,n1,n2,n3),n4) by Th25
    .=crossover(crossover(p1,p2,n2,n1,n3),crossover(p2,p1,n2,n1,n3),n4) by Th25
;
  crossover(p1,p2,n1,n2,n3,n4) =crossover(crossover(p1,p2,n2,n1,n3),
  crossover(p2,p1,n1,n2,n3),n4) by Th25
    .=crossover(crossover(p1,p2,n2,n1,n3),crossover(p2,p1,n2,n1,n3),n4) by Th25
    .=crossover(crossover(p1,p2,n2,n3,n1),crossover(p2,p1,n2,n1,n3),n4) by Th25
    .=crossover(crossover(p1,p2,n2,n3,n1),crossover(p2,p1,n2,n3,n1),n4) by Th25
;
  hence thesis by A2,A15,A1,A8,A4,A6,A3,A10,A14,A12,A9,A13,A11,A7,A5;
end;
