:: GENEALG1 semantic presentation

begin


theorem Th1: :: GENEALG1:1
for D being non empty set
for f1, f2 being FinSequence of D
for n being Nat st n <= len f1 holds
(f1 ^ f2) /^ n = (f1 /^ n) ^ f2
proof
let D be non empty set ; ::_thesis: for f1, f2 being FinSequence of D
for n being Nat st n <= len f1 holds
(f1 ^ f2) /^ n = (f1 /^ n) ^ f2
let f1, f2 be FinSequence of D; ::_thesis: for n being Nat st n <= len f1 holds
(f1 ^ f2) /^ n = (f1 /^ n) ^ f2
let n be Nat; ::_thesis: ( n <= len f1 implies (f1 ^ f2) /^ n = (f1 /^ n) ^ f2 )
assume A1: n <= len f1 ; ::_thesis: (f1 ^ f2) /^ n = (f1 /^ n) ^ f2
reconsider n = n as Element of NAT by ORDINAL1:def_12;
 len (f1 ^ f2) = (len f1) + (len f2) by FINSEQ_1:22;
then  len f1 <= len (f1 ^ f2) by NAT_1:11;
then A2: n <= len (f1 ^ f2) by A1, XXREAL_0:2;
then A3: len ((f1 ^ f2) /^ n) = (len (f1 ^ f2)) - n by RFINSEQ:def_1;
A4: len ((f1 /^ n) ^ f2) = (len (f1 /^ n)) + (len f2) by FINSEQ_1:22
.= ((len f1) - n) + (len f2) by A1, RFINSEQ:def_1
.= ((len f1) + (len f2)) - n ;
A5: for i being Nat st 1 <= i & i <= len ((f1 ^ f2) /^ n) holds
((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len ((f1 ^ f2) /^ n) implies ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i )
assume that
A6: 1 <= i and
A7: i <= len ((f1 ^ f2) /^ n) ; ::_thesis: ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i
 i in Seg (len ((f1 ^ f2) /^ n)) by A6, A7, FINSEQ_1:1;
then A8: i in dom ((f1 ^ f2) /^ n) by FINSEQ_1:def_3;
reconsider i = i as Element of NAT by ORDINAL1:def_12;
now__::_thesis:_((f1_^_f2)_/^_n)_._i_=_((f1_/^_n)_^_f2)_._i
percases ( i <= (len f1) - n or (len f1) - n < i ) ;
supposeA9: i <= (len f1) - n ; ::_thesis: ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i
 i <= i + n by NAT_1:11;
then A10: 1 <= i + n by A6, XXREAL_0:2;
 i + n <= len f1 by A9, XREAL_1:19;
then  i + n in Seg (len f1) by A10, FINSEQ_1:1;
then A11: i + n in dom f1 by FINSEQ_1:def_3;
 i <= len (f1 /^ n) by A1, A9, RFINSEQ:def_1;
then  i in Seg (len (f1 /^ n)) by A6, FINSEQ_1:1;
then A12: i in dom (f1 /^ n) by FINSEQ_1:def_3;
then A13: ((f1 /^ n) ^ f2) . i = (f1 /^ n) . i by FINSEQ_1:def_7
.= f1 . (i + n) by A1, A12, RFINSEQ:def_1 ;
((f1 ^ f2) /^ n) . i = (f1 ^ f2) . (i + n) by A2, A8, RFINSEQ:def_1
.= f1 . (i + n) by A11, FINSEQ_1:def_7 ;
hence  ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i by A13; ::_thesis: verum
end;
supposeA14: (len f1) - n < i ; ::_thesis: ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i
then A15: len (f1 /^ n) < i by A1, RFINSEQ:def_1;
 i <= len ((f1 /^ n) ^ f2) by A3, A4, A7, FINSEQ_1:22;
then A16: ((f1 /^ n) ^ f2) . i = f2 . (i - (len (f1 /^ n))) by A15, FINSEQ_1:24
.= f2 . (i - ((len f1) - n)) by A1, RFINSEQ:def_1
.= f2 . ((i + n) - (len f1)) ;
A17: i + n <= len (f1 ^ f2) by A3, A7, XREAL_1:19;
A18: len f1 < i + n by A14, XREAL_1:19;
((f1 ^ f2) /^ n) . i = (f1 ^ f2) . (i + n) by A2, A8, RFINSEQ:def_1
.= f2 . ((i + n) - (len f1)) by A18, A17, FINSEQ_1:24 ;
hence  ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i by A16; ::_thesis: verum
end;
end;
end;
hence  ((f1 ^ f2) /^ n) . i = ((f1 /^ n) ^ f2) . i ; ::_thesis: verum
end;
 len ((f1 ^ f2) /^ n) = len ((f1 /^ n) ^ f2) by A3, A4, FINSEQ_1:22;
hence  (f1 ^ f2) /^ n = (f1 /^ n) ^ f2 by A5, FINSEQ_1:14; ::_thesis: verum
end;

theorem Th2: :: GENEALG1:2
for D being non empty set
for f1, f2 being FinSequence of D
for i being Element of NAT holds (f1 ^ f2) | ((len f1) + i) = f1 ^ (f2 | i)
proof
let D be non empty set ; ::_thesis: for f1, f2 being FinSequence of D
for i being Element of NAT holds (f1 ^ f2) | ((len f1) + i) = f1 ^ (f2 | i)
let f1, f2 be FinSequence of D; ::_thesis: for i being Element of NAT holds (f1 ^ f2) | ((len f1) + i) = f1 ^ (f2 | i)
let i be Element of NAT ; ::_thesis: (f1 ^ f2) | ((len f1) + i) = f1 ^ (f2 | i)
 ( (f1 ^ f2) | ((len f1) + i) = (f1 ^ f2) | (Seg ((len f1) + i)) & f2 | i = f2 | (Seg i) ) by FINSEQ_1:def_15;
hence  (f1 ^ f2) | ((len f1) + i) = f1 ^ (f2 | i) by FINSEQ_6:14; ::_thesis: verum
end;

definition
mode Gene-Set is non empty non-empty FinSequence;
end;

notation
let S be Gene-Set;
synonym  GA-Space S for  Union S;
end;

registration
let f be non empty non-empty Function;
cluster GA-Space f ->  non empty ;
coherence
 not Union f is empty
proof
 Union f = union (rng f) by CARD_3:def_4;
hence  not Union f is empty ; ::_thesis: verum
end;
end;

definition
let S be Gene-Set;
mode Individual of S ->  FinSequence of GA-Space S means :Def1: :: GENEALG1:def 1
( len it = len S & ( for i being Element of NAT st i in dom it holds
it . i in S . i ) );
existence
 ex b1 being FinSequence of GA-Space S st
( len b1 = len S & ( for i being Element of NAT st i in dom b1 holds
b1 . i in S . i ) )
proof
defpred S1[ set , set ] means $2 in S . $1;
A1: for k being Nat st k in Seg (len S) holds
ex x being Element of GA-Space S st S1[k,x]
proof
let k be Nat; ::_thesis: ( k in Seg (len S) implies ex x being Element of GA-Space S st S1[k,x] )
assume A2: k in Seg (len S) ; ::_thesis: ex x being Element of GA-Space S st S1[k,x]
then reconsider k9 = k as Element of dom S by FINSEQ_1:def_3;
 S . k9 <> {} ;
then consider x being Element of S . k such that
A3: x in [#] (S . k) by SUBSET_1:4;
 k in dom S by A2, FINSEQ_1:def_3;
then  S . k in rng S by FUNCT_1:def_3;
then  [#] (S . k) c= union (rng S) by ZFMISC_1:74;
then reconsider x = x as Element of GA-Space S by A3, CARD_3:def_4;
take  x ; ::_thesis: S1[k,x]
thus  S1[k,x] by A3; ::_thesis: verum
end;
consider IT being FinSequence of GA-Space S such that
A4: ( dom IT = Seg (len S) & ( for k being Nat st k in Seg (len S) holds
S1[k,IT . k] ) ) from FINSEQ_1:sch_5(A1);
take  IT ; ::_thesis: ( len IT = len S & ( for i being Element of NAT st i in dom IT holds
IT . i in S . i ) )
thus  ( len IT = len S & ( for i being Element of NAT st i in dom IT holds
IT . i in S . i ) ) by A4, FINSEQ_1:def_3; ::_thesis: verum
end;
end;

:: deftheorem Def1 defines Individual GENEALG1:def_1_:_
 for S being Gene-Set
for b2 being FinSequence of GA-Space S holds
( b2 is Individual of S iff ( len b2 = len S & ( for i being Element of NAT st i in dom b2 holds
b2 . i in S . i ) ) );

begin

definition
let S be Gene-Set;
let p1, p2 be FinSequence of GA-Space S;
let n be Element of NAT ;
func crossover (p1,p2,n) ->  FinSequence of GA-Space S equals :: GENEALG1:def 2
(p1 | n) ^ (p2 /^ n);
correctness
coherence
(p1 | n) ^ (p2 /^ n) is FinSequence of GA-Space S;
;
end;

:: deftheorem  defines crossover GENEALG1:def_2_:_
 for S being Gene-Set
for p1, p2 being FinSequence of GA-Space S
for n being Element of NAT holds crossover (p1,p2,n) = (p1 | n) ^ (p2 /^ n);

definition
let S be Gene-Set;
let p1, p2 be FinSequence of GA-Space S;
let n1, n2 be Element of NAT ;
func crossover (p1,p2,n1,n2) ->  FinSequence of GA-Space S equals :: GENEALG1:def 3
 crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n2);
correctness
coherence
crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n2) is FinSequence of GA-Space S;
;
end;

:: deftheorem  defines crossover GENEALG1:def_3_:_
 for S being Gene-Set
for p1, p2 being FinSequence of GA-Space S
for n1, n2 being Element of NAT holds crossover (p1,p2,n1,n2) = crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n2);

definition
let S be Gene-Set;
let p1, p2 be FinSequence of GA-Space S;
let n1, n2, n3 be Element of NAT ;
func crossover (p1,p2,n1,n2,n3) ->  FinSequence of GA-Space S equals :: GENEALG1:def 4
 crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n3);
correctness
coherence
crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n3) is FinSequence of GA-Space S;
;
end;

:: deftheorem  defines crossover GENEALG1:def_4_:_
 for S being Gene-Set
for p1, p2 being FinSequence of GA-Space S
for n1, n2, n3 being Element of NAT holds crossover (p1,p2,n1,n2,n3) = crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n3);

definition
let S be Gene-Set;
let p1, p2 be FinSequence of GA-Space S;
let n1, n2, n3, n4 be Element of NAT ;
func crossover (p1,p2,n1,n2,n3,n4) ->  FinSequence of GA-Space S equals :: GENEALG1:def 5
 crossover ((crossover (p1,p2,n1,n2,n3)),(crossover (p2,p1,n1,n2,n3)),n4);
correctness
coherence
crossover ((crossover (p1,p2,n1,n2,n3)),(crossover (p2,p1,n1,n2,n3)),n4) is FinSequence of GA-Space S;
;
end;

:: deftheorem  defines crossover GENEALG1:def_5_:_
 for S being Gene-Set
for p1, p2 being FinSequence of GA-Space S
for n1, n2, n3, n4 being Element of NAT holds crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n1,n2,n3)),(crossover (p2,p1,n1,n2,n3)),n4);

definition
let S be Gene-Set;
let p1, p2 be FinSequence of GA-Space S;
let n1, n2, n3, n4, n5 be Element of NAT ;
func crossover (p1,p2,n1,n2,n3,n4,n5) ->  FinSequence of GA-Space S equals :: GENEALG1:def 6
 crossover ((crossover (p1,p2,n1,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5);
correctness
coherence
crossover ((crossover (p1,p2,n1,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5) is FinSequence of GA-Space S;
;
end;

:: deftheorem  defines crossover GENEALG1:def_6_:_
 for S being Gene-Set
for p1, p2 being FinSequence of GA-Space S
for n1, n2, n3, n4, n5 being Element of NAT holds crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n1,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5);

definition
let S be Gene-Set;
let p1, p2 be FinSequence of GA-Space S;
let n1, n2, n3, n4, n5, n6 be Element of NAT ;
func crossover (p1,p2,n1,n2,n3,n4,n5,n6) ->  FinSequence of GA-Space S equals :: GENEALG1:def 7
 crossover ((crossover (p1,p2,n1,n2,n3,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6);
correctness
coherence
crossover ((crossover (p1,p2,n1,n2,n3,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) is FinSequence of GA-Space S;
;
end;

:: deftheorem  defines crossover GENEALG1:def_7_:_
 for S being Gene-Set
for p1, p2 being FinSequence of GA-Space S
for n1, n2, n3, n4, n5, n6 being Element of NAT holds crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n1,n2,n3,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6);

begin


theorem Th3: :: GENEALG1:3
for n being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n) is Individual of S
proof
let n be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n) is Individual of S
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n) is Individual of S
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n) is Individual of S
A1: len (crossover (p1,p2,n)) = len S
proof
A2: len (crossover (p1,p2,n)) = (len (p1 | n)) + (len (p2 /^ n)) by FINSEQ_1:22;
now__::_thesis:_len_(crossover_(p1,p2,n))_=_len_S
percases ( n <= len p1 or n > len p1 ) ;
supposeA3: n <= len p1 ; ::_thesis: len (crossover (p1,p2,n)) = len S
len (p2 /^ n) = (len p2) -' n by RFINSEQ:29
.= (len S) -' n by Def1
.= (len p1) -' n by Def1
.= (len p1) - n by A3, XREAL_1:233 ;
then  len (crossover (p1,p2,n)) = n + ((len p1) - n) by A2, A3, FINSEQ_1:59
.= len p1 ;
hence  len (crossover (p1,p2,n)) = len S by Def1; ::_thesis: verum
end;
supposeA4: n > len p1 ; ::_thesis: len (crossover (p1,p2,n)) = len S
 p1 | n = p1 | (Seg n) by FINSEQ_1:def_15;
then A5: p1 | n = p1 by A4, FINSEQ_2:20;
A6: (len p1) - n < 0 by A4, XREAL_1:49;
len (p2 /^ n) = (len p2) -' n by RFINSEQ:29
.= (len S) -' n by Def1
.= (len p1) -' n by Def1
.= 0 by A6, XREAL_0:def_2 ;
hence  len (crossover (p1,p2,n)) = len S by A2, A5, Def1; ::_thesis: verum
end;
end;
end;
hence  len (crossover (p1,p2,n)) = len S ; ::_thesis: verum
end;
 for i being Element of NAT st i in dom (crossover (p1,p2,n)) holds
(crossover (p1,p2,n)) . i in S . i
proof
let i be Element of NAT ; ::_thesis: ( i in dom (crossover (p1,p2,n)) implies (crossover (p1,p2,n)) . i in S . i )
assume A7: i in dom (crossover (p1,p2,n)) ; ::_thesis: (crossover (p1,p2,n)) . i in S . i
now__::_thesis:_(crossover_(p1,p2,n))_._i_in_S_._i
percases ( i in dom (p1 | n) or ex k being Nat st
( k in dom (p2 /^ n) & i = (len (p1 | n)) + k ) ) by A7, FINSEQ_1:25;
supposeA8: i in dom (p1 | n) ; ::_thesis: (crossover (p1,p2,n)) . i in S . i
A9: dom (p1 | n) c= dom p1 by FINSEQ_5:18;
(p1 | n) . i = (p1 | n) /. i by A8, PARTFUN1:def_6
.= p1 /. i by A8, FINSEQ_4:70
.= p1 . i by A8, A9, PARTFUN1:def_6 ;
then  (p1 | n) . i in S . i by A8, A9, Def1;
hence  (crossover (p1,p2,n)) . i in S . i by A8, FINSEQ_1:def_7; ::_thesis: verum
end;
suppose ex k being Nat st
( k in dom (p2 /^ n) & i = (len (p1 | n)) + k ) ; ::_thesis: (crossover (p1,p2,n)) . i in S . i
then consider k being Nat such that
A10: k in dom (p2 /^ n) and
A11: i = (len (p1 | n)) + k ;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
A12: n <= len (p1 | n)
proof
 n + k in dom p2 by A10, FINSEQ_5:26;
then  n + k in Seg (len p2) by FINSEQ_1:def_3;
then  n + k <= len p2 by FINSEQ_1:1;
then  n + k <= len S by Def1;
then  n + k <= len p1 by Def1;
then A13: k <= (len p1) - n by XREAL_1:19;
 k in Seg (len (p2 /^ n)) by A10, FINSEQ_1:def_3;
then  1 <= k by FINSEQ_1:1;
then  1 <= (len p1) - n by A13, XXREAL_0:2;
then  1 + n <= len p1 by XREAL_1:19;
then A14: n <= (len p1) - 1 by XREAL_1:19;
 len p1 <= (len p1) + 1 by NAT_1:11;
then A15: (len p1) - 1 <= len p1 by XREAL_1:20;
assume  n > len (p1 | n) ; ::_thesis: contradiction
hence  contradiction by A14, A15, FINSEQ_1:59, XXREAL_0:2; ::_thesis: verum
end;
 len (p1 | n) <= n by FINSEQ_5:17;
then A16: len (p1 | n) = n by A12, XXREAL_0:1;
then  n + k in Seg (len S) by A1, A7, A11, FINSEQ_1:def_3;
then  n + k in Seg (len p2) by Def1;
then A17: n + k in dom p2 by FINSEQ_1:def_3;
(crossover (p1,p2,n)) . i = (p2 /^ n) . k by A10, A11, FINSEQ_1:def_7
.= (p2 /^ n) /. k by A10, PARTFUN1:def_6
.= p2 /. (n + k) by A10, FINSEQ_5:27
.= p2 . (n + k) by A17, PARTFUN1:def_6 ;
hence  (crossover (p1,p2,n)) . i in S . i by A11, A16, A17, Def1; ::_thesis: verum
end;
end;
end;
hence  (crossover (p1,p2,n)) . i in S . i ; ::_thesis: verum
end;
hence  crossover (p1,p2,n) is Individual of S by A1, Def1; ::_thesis: verum
end;

definition
let S be Gene-Set;
let p1, p2 be Individual of S;
let n be Element of NAT ;
:: original: crossover
redefine func crossover (p1,p2,n) ->  Individual of S;
correctness
coherence
crossover (p1,p2,n) is Individual of S;
by Th3;
end;

theorem Th4: :: GENEALG1:4
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,0) = p2
proof
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,0) = p2
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,0) = p2
crossover (p1,p2,0) = p2 /^ 0 by FINSEQ_1:34
.= p2 by FINSEQ_5:28 ;
hence  crossover (p1,p2,0) = p2 ; ::_thesis: verum
end;

theorem Th5: :: GENEALG1:5
for n being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n >= len p1 holds
crossover (p1,p2,n) = p1
proof
let n be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S st n >= len p1 holds
crossover (p1,p2,n) = p1
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n >= len p1 holds
crossover (p1,p2,n) = p1
let p1, p2 be Individual of S; ::_thesis: ( n >= len p1 implies crossover (p1,p2,n) = p1 )
assume A1: n >= len p1 ; ::_thesis: crossover (p1,p2,n) = p1
then  n >= len S by Def1;
then A2: n >= len p2 by Def1;
crossover (p1,p2,n) = p1 ^ (p2 /^ n) by A1, FINSEQ_1:58
.= p1 ^ {} by A2, FINSEQ_5:32
.= p1 by FINSEQ_1:34 ;
hence  crossover (p1,p2,n) = p1 ; ::_thesis: verum
end;

begin

theorem Th6: :: GENEALG1:6
for n1, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2) is Individual of S
proof
let n1, n2 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2) is Individual of S
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2) is Individual of S
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n2) is Individual of S
reconsider q1 = crossover (p1,p2,n1), q2 = crossover (p2,p1,n1) as Individual of S ;
 crossover (q1,q2,n2) is Individual of S ;
hence  crossover (p1,p2,n1,n2) is Individual of S ; ::_thesis: verum
end;

definition
let S be Gene-Set;
let p1, p2 be Individual of S;
let n1, n2 be Element of NAT ;
:: original: crossover
redefine func crossover (p1,p2,n1,n2) ->  Individual of S;
correctness
coherence
crossover (p1,p2,n1,n2) is Individual of S;
by Th6;
end;

theorem Th7: :: GENEALG1:7
for n being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,0,n) = crossover (p2,p1,n)
proof
let n be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,0,n) = crossover (p2,p1,n)
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,0,n) = crossover (p2,p1,n)
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,0,n) = crossover (p2,p1,n)
crossover (p1,p2,0,n) = crossover (p2,(crossover (p2,p1,0)),n) by Th4
.= crossover (p2,p1,n) by Th4 ;
hence  crossover (p1,p2,0,n) = crossover (p2,p1,n) ; ::_thesis: verum
end;

theorem Th8: :: GENEALG1:8
for n being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n,0) = crossover (p2,p1,n)
proof
let n be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n,0) = crossover (p2,p1,n)
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n,0) = crossover (p2,p1,n)
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n,0) = crossover (p2,p1,n)
reconsider q1 = crossover (p1,p2,n) as Individual of S ;
reconsider q2 = crossover (p2,p1,n) as Individual of S ;
crossover (p1,p2,n,0) = crossover (q1,q2,0)
.= q2 by Th4 ;
hence  crossover (p1,p2,n,0) = crossover (p2,p1,n) ; ::_thesis: verum
end;

theorem Th9: :: GENEALG1:9
for n1, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 holds
crossover (p1,p2,n1,n2) = crossover (p1,p2,n2)
proof
let n1, n2 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 holds
crossover (p1,p2,n1,n2) = crossover (p1,p2,n2)
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n1 >= len p1 holds
crossover (p1,p2,n1,n2) = crossover (p1,p2,n2)
let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 implies crossover (p1,p2,n1,n2) = crossover (p1,p2,n2) )
assume A1: n1 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2)
then  n1 >= len S by Def1;
then A2: n1 >= len p2 by Def1;
 crossover (p1,p2,n1,n2) = crossover (p1,(crossover (p2,p1,n1)),n2) by A1, Th5;
hence  crossover (p1,p2,n1,n2) = crossover (p1,p2,n2) by A2, Th5; ::_thesis: verum
end;

theorem Th10: :: GENEALG1:10
for n2, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n2 >= len p1 holds
crossover (p1,p2,n1,n2) = crossover (p1,p2,n1)
proof
let n2, n1 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S st n2 >= len p1 holds
crossover (p1,p2,n1,n2) = crossover (p1,p2,n1)
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n2 >= len p1 holds
crossover (p1,p2,n1,n2) = crossover (p1,p2,n1)
let p1, p2 be Individual of S; ::_thesis: ( n2 >= len p1 implies crossover (p1,p2,n1,n2) = crossover (p1,p2,n1) )
assume  n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n1)
then  n2 >= len S by Def1;
then A1: n2 >= len (crossover (p1,p2,n1)) by Def1;
 crossover (p1,p2,n1,n2) = crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n2) ;
hence  crossover (p1,p2,n1,n2) = crossover (p1,p2,n1) by A1, Th5; ::_thesis: verum
end;

theorem :: GENEALG1:11
for n1, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 holds
crossover (p1,p2,n1,n2) = p1
proof
let n1, n2 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 holds
crossover (p1,p2,n1,n2) = p1
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 holds
crossover (p1,p2,n1,n2) = p1
let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2) = p1 )
assume that
A1: n1 >= len p1 and
A2: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2) = p1
 crossover (p1,p2,n1,n2) = crossover (p1,p2,n2) by A1, Th9;
hence  crossover (p1,p2,n1,n2) = p1 by A2, Th5; ::_thesis: verum
end;

theorem Th12: :: GENEALG1:12
for n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n1,n1) = p1
proof
let n1 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n1,n1) = p1
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n1) = p1
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n1) = p1
A1: ((p2 | n1) ^ (p1 /^ n1)) /^ n1 = p1 /^ n1
proof
now__::_thesis:_((p2_|_n1)_^_(p1_/^_n1))_/^_n1_=_p1_/^_n1
percases ( n1 <= len p2 or n1 > len p2 ) ;
suppose n1 <= len p2 ; ::_thesis: ((p2 | n1) ^ (p1 /^ n1)) /^ n1 = p1 /^ n1
then  len (p2 | n1) = n1 by FINSEQ_1:59;
hence  ((p2 | n1) ^ (p1 /^ n1)) /^ n1 = p1 /^ n1 by FINSEQ_5:37; ::_thesis: verum
end;
suppose n1 > len p2 ; ::_thesis: ((p2 | n1) ^ (p1 /^ n1)) /^ n1 = p1 /^ n1
then A2: n1 > len S by Def1;
then  n1 > len (crossover (p2,p1,n1)) by Def1;
then A3: ((p2 | n1) ^ (p1 /^ n1)) /^ n1 = {} by FINSEQ_5:32;
 n1 > len p1 by A2, Def1;
hence  ((p2 | n1) ^ (p1 /^ n1)) /^ n1 = p1 /^ n1 by A3, FINSEQ_5:32; ::_thesis: verum
end;
end;
end;
hence  ((p2 | n1) ^ (p1 /^ n1)) /^ n1 = p1 /^ n1 ; ::_thesis: verum
end;
 ((p1 | n1) ^ (p2 /^ n1)) | n1 = p1 | n1
proof
now__::_thesis:_((p1_|_n1)_^_(p2_/^_n1))_|_n1_=_p1_|_n1
percases ( n1 <= len p1 or n1 > len p1 ) ;
suppose n1 <= len p1 ; ::_thesis: ((p1 | n1) ^ (p2 /^ n1)) | n1 = p1 | n1
then  len (p1 | n1) = n1 by FINSEQ_1:59;
hence  ((p1 | n1) ^ (p2 /^ n1)) | n1 = p1 | n1 by FINSEQ_5:23; ::_thesis: verum
end;
supposeA4: n1 > len p1 ; ::_thesis: ((p1 | n1) ^ (p2 /^ n1)) | n1 = p1 | n1
then A5: n1 > len S by Def1;
then A6: n1 > len p2 by Def1;
 n1 > len (crossover (p1,p2,n1)) by A5, Def1;
then ((p1 | n1) ^ (p2 /^ n1)) | n1 = (p1 | n1) ^ (p2 /^ n1) by FINSEQ_1:58
.= p1 ^ (p2 /^ n1) by A4, FINSEQ_1:58
.= p1 ^ {} by A6, FINSEQ_5:32
.= p1 by FINSEQ_1:34
.= p1 | n1 by A4, FINSEQ_1:58 ;
hence  ((p1 | n1) ^ (p2 /^ n1)) | n1 = p1 | n1 ; ::_thesis: verum
end;
end;
end;
hence  ((p1 | n1) ^ (p2 /^ n1)) | n1 = p1 | n1 ; ::_thesis: verum
end;
hence  crossover (p1,p2,n1,n1) = p1 by A1, RFINSEQ:8; ::_thesis: verum
end;

theorem Th13: :: GENEALG1:13
for n1, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1)
proof
let n1, n2 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1)
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1)
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1)
A1: len p1 = len S by Def1;
 len ((p2 | n2) ^ (p1 /^ n2)) = len (crossover (p2,p1,n2)) ;
then A2: len ((p2 | n2) ^ (p1 /^ n2)) = len S by Def1;
A3: len p2 = len S by Def1;
 len ((p1 | n2) ^ (p2 /^ n2)) = len (crossover (p1,p2,n2)) ;
then A4: len ((p1 | n2) ^ (p2 /^ n2)) = len S by Def1;
now__::_thesis:_crossover_(p1,p2,n1,n2)_=_crossover_(p1,p2,n2,n1)
percases ( ( n1 >= n2 & n2 > 0 ) or ( n1 < n2 & n2 > 0 ) or n2 = 0 ) by NAT_1:3;
supposeA5: ( n1 >= n2 & n2 > 0 ) ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1)
now__::_thesis:_crossover_(p1,p2,n1,n2)_=_crossover_(p1,p2,n2,n1)
percases ( n1 >= len p1 or n1 < len p1 ) ;
supposeA6: n1 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1)
then  p2 | n1 = p2 by A1, A3, FINSEQ_1:58;
then A7: (p2 | n1) ^ (p1 /^ n1) = p2 ^ {} by A6, FINSEQ_5:32
.= p2 by FINSEQ_1:34 ;
 p1 | n1 = p1 by A6, FINSEQ_1:58;
then A8: (p1 | n1) ^ (p2 /^ n1) = p1 ^ {} by A1, A3, A6, FINSEQ_5:32
.= p1 by FINSEQ_1:34 ;
 ((p1 | n2) ^ (p2 /^ n2)) | n1 = (p1 | n2) ^ (p2 /^ n2) by A1, A4, A6, FINSEQ_1:58;
then  crossover (p1,p2,n2,n1) = ((p1 | n2) ^ (p2 /^ n2)) ^ {} by A1, A2, A6, FINSEQ_5:32
.= (p1 | n2) ^ (p2 /^ n2) by FINSEQ_1:34 ;
hence  crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) by A8, A7; ::_thesis: verum
end;
supposeA9: n1 < len p1 ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1)
then  len (p1 | n1) = n1 by FINSEQ_1:59;
then A10: ((p1 | n1) ^ (p2 /^ n1)) | n2 = (p1 | n1) | n2 by A5, FINSEQ_5:22
.= p1 | n2 by A5, FINSEQ_5:77 ;
 n1 <= len p2 by A3, A9, Def1;
then  n2 <= len (p2 | n1) by A5, FINSEQ_1:59;
then A11: crossover (p1,p2,n1,n2) = (p1 | n2) ^ (((p2 | n1) /^ n2) ^ (p1 /^ n1)) by A10, Th1
.= ((p1 | n2) ^ ((p2 | n1) /^ n2)) ^ (p1 /^ n1) by FINSEQ_1:32 ;
 len (p2 | n2) = n2 by A1, A3, A5, A9, FINSEQ_1:59, XXREAL_0:2;
then consider i being Nat such that
A12: (len (p2 | n2)) + i = n1 by A5, NAT_1:10;
reconsider i = i as Element of NAT by ORDINAL1:def_12;
A13: len (p1 | n2) = n2 by A5, A9, FINSEQ_1:59, XXREAL_0:2;
then A14: (len (p1 | n2)) + i = n1 by A1, A3, A5, A9, A12, FINSEQ_1:59, XXREAL_0:2;
then  i = n1 - n2 by A13;
then A15: i = n1 -' n2 by A5, XREAL_1:233;
A16: ((p1 | n2) ^ (p2 /^ n2)) | n1 = (p1 | n2) ^ ((p2 /^ n2) | i) by A14, Th2
.= (p1 | n2) ^ ((p2 | n1) /^ n2) by A15, FINSEQ_5:80 ;
((p2 | n2) ^ (p1 /^ n2)) /^ n1 = (p1 /^ n2) /^ i by A12, FINSEQ_5:36
.= p1 /^ (n2 + i) by FINSEQ_6:81
.= p1 /^ n1 by A1, A3, A5, A9, A12, FINSEQ_1:59, XXREAL_0:2 ;
hence  crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) by A11, A16; ::_thesis: verum
end;
end;
end;
hence  crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) ; ::_thesis: verum
end;
supposeA17: ( n1 < n2 & n2 > 0 ) ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1)
now__::_thesis:_crossover_(p1,p2,n1,n2)_=_crossover_(p1,p2,n2,n1)
percases ( n1 >= len p1 or n1 < len p1 ) ;
supposeA18: n1 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1)
then  n2 >= len p1 by A17, XXREAL_0:2;
then A19: n2 >= len p2 by A3, Def1;
A20: n1 >= len p2 by A3, A18, Def1;
then  p2 | n1 = p2 by FINSEQ_1:58;
then (p2 | n1) ^ (p1 /^ n1) = p2 ^ {} by A18, FINSEQ_5:32
.= p2 by FINSEQ_1:34 ;
then A21: ((p2 | n1) ^ (p1 /^ n1)) /^ n2 = {} by A19, FINSEQ_5:32;
 p1 | n2 = p1 by A17, A18, FINSEQ_1:58, XXREAL_0:2;
then (p1 | n2) ^ (p2 /^ n2) = p1 ^ {} by A19, FINSEQ_5:32
.= p1 by FINSEQ_1:34 ;
then A22: ((p1 | n2) ^ (p2 /^ n2)) | n1 = p1 by A18, FINSEQ_1:58;
 p2 | n2 = p2 by A19, FINSEQ_1:58;
then (p2 | n2) ^ (p1 /^ n2) = p2 ^ {} by A17, A18, FINSEQ_5:32, XXREAL_0:2
.= p2 by FINSEQ_1:34 ;
then A23: ((p2 | n2) ^ (p1 /^ n2)) /^ n1 = {} by A20, FINSEQ_5:32;
 p1 | n1 = p1 by A18, FINSEQ_1:58;
then (p1 | n1) ^ (p2 /^ n1) = p1 ^ {} by A20, FINSEQ_5:32
.= p1 by FINSEQ_1:34 ;
hence  crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) by A17, A18, A21, A22, A23, FINSEQ_1:58, XXREAL_0:2; ::_thesis: verum
end;
supposeA24: n1 < len p1 ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1)
 n1 <= len (p1 | n2)
proof
now__::_thesis:_n1_<=_len_(p1_|_n2)
percases ( n2 >= len p1 or n2 < len p1 ) ;
suppose n2 >= len p1 ; ::_thesis: n1 <= len (p1 | n2)
hence  n1 <= len (p1 | n2) by A24, FINSEQ_1:58; ::_thesis: verum
end;
suppose n2 < len p1 ; ::_thesis: n1 <= len (p1 | n2)
hence  n1 <= len (p1 | n2) by A17, FINSEQ_1:59; ::_thesis: verum
end;
end;
end;
hence  n1 <= len (p1 | n2) ; ::_thesis: verum
end;
then A25: ((p1 | n2) ^ (p2 /^ n2)) | n1 = (p1 | n2) | n1 by FINSEQ_5:22
.= p1 | n1 by A17, FINSEQ_5:77 ;
A26: len (p1 | n1) = n1 by A24, FINSEQ_1:59;
then consider i being Nat such that
A27: (len (p1 | n1)) + i = n2 by A17, NAT_1:10;
reconsider i = i as Element of NAT by ORDINAL1:def_12;
A28: ((p1 | n1) ^ (p2 /^ n1)) | n2 = (p1 | n1) ^ ((p2 /^ n1) | i) by A27, Th2;
 len (p2 | n1) = n1 by A1, A3, A24, FINSEQ_1:59;
then  (len (p2 | n1)) + i = n2 by A24, A27, FINSEQ_1:59;
then A29: crossover (p1,p2,n1,n2) = ((p1 | n1) ^ ((p2 /^ n1) | i)) ^ ((p1 /^ n1) /^ i) by A28, FINSEQ_5:36
.= ((p1 | n1) ^ ((p2 /^ n1) | i)) ^ (p1 /^ (n1 + i)) by FINSEQ_6:81
.= ((p1 | n1) ^ ((p2 /^ n1) | i)) ^ (p1 /^ n2) by A24, A27, FINSEQ_1:59 ;
A30: i = n2 - n1 by A26, A27;
 n1 <= len (p2 | n2)
proof
now__::_thesis:_n1_<=_len_(p2_|_n2)
percases ( n2 >= len p2 or n2 < len p2 ) ;
suppose n2 >= len p2 ; ::_thesis: n1 <= len (p2 | n2)
hence  n1 <= len (p2 | n2) by A1, A3, A24, FINSEQ_1:58; ::_thesis: verum
end;
suppose n2 < len p2 ; ::_thesis: n1 <= len (p2 | n2)
hence  n1 <= len (p2 | n2) by A17, FINSEQ_1:59; ::_thesis: verum
end;
end;
end;
hence  n1 <= len (p2 | n2) ; ::_thesis: verum
end;
then ((p2 | n2) ^ (p1 /^ n2)) /^ n1 = ((p2 | n2) /^ n1) ^ (p1 /^ n2) by Th1
.= ((p2 /^ n1) | (n2 -' n1)) ^ (p1 /^ n2) by FINSEQ_5:80
.= ((p2 /^ n1) | i) ^ (p1 /^ n2) by A17, A30, XREAL_1:233 ;
hence  crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) by A29, A25, FINSEQ_1:32; ::_thesis: verum
end;
end;
end;
hence  crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) ; ::_thesis: verum
end;
supposeA31: n2 = 0 ; ::_thesis: crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1)
then  crossover (p1,p2,n1,n2) = crossover (p2,p1,n1) by Th8
.= crossover (p1,p2,0,n1) by Th7 ;
hence  crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) by A31; ::_thesis: verum
end;
end;
end;
hence  crossover (p1,p2,n1,n2) = crossover (p1,p2,n2,n1) ; ::_thesis: verum
end;

begin

theorem Th14: :: GENEALG1:14
for n1, n2, n3 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3) is Individual of S
proof
let n1, n2, n3 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3) is Individual of S
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3) is Individual of S
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n2,n3) is Individual of S
reconsider q1 = crossover (p1,p2,n1,n2), q2 = crossover (p2,p1,n1,n2) as Individual of S ;
 crossover (q1,q2,n3) is Individual of S ;
hence  crossover (p1,p2,n1,n2,n3) is Individual of S ; ::_thesis: verum
end;

definition
let S be Gene-Set;
let p1, p2 be Individual of S;
let n1, n2, n3 be Element of NAT ;
:: original: crossover
redefine func crossover (p1,p2,n1,n2,n3) ->  Individual of S;
correctness
coherence
crossover (p1,p2,n1,n2,n3) is Individual of S;
by Th14;
end;

theorem Th15: :: GENEALG1:15
for n2, n3, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,n3) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2) )
proof
let n2, n3, n1 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,n3) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,n3) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,0,n2,n3) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,n3) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2) )
crossover (p1,p2,0,n2,n3) = crossover ((crossover (p2,p1,n2)),(crossover (p2,p1,0,n2)),n3) by Th7
.= crossover ((crossover (p2,p1,n2)),(crossover (p1,p2,n2)),n3) by Th7 ;
hence  crossover (p1,p2,0,n2,n3) = crossover (p2,p1,n2,n3) ; ::_thesis: ( crossover (p1,p2,n1,0,n3) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2) )
crossover (p1,p2,n1,0,n3) = crossover ((crossover (p2,p1,n1)),(crossover (p2,p1,n1,0)),n3) by Th8
.= crossover ((crossover (p2,p1,n1)),(crossover (p1,p2,n1)),n3) by Th8 ;
hence  crossover (p1,p2,n1,0,n3) = crossover (p2,p1,n1,n3) ; ::_thesis: crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2)
 crossover (p1,p2,n1,n2,0) = crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),0) ;
hence  crossover (p1,p2,n1,n2,0) = crossover (p2,p1,n1,n2) by Th4; ::_thesis: verum
end;

theorem :: GENEALG1:16
for n3, n1, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,n3) = crossover (p1,p2,n3) & crossover (p1,p2,n1,0,0) = crossover (p1,p2,n1) & crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2) )
proof
let n3, n1, n2 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,n3) = crossover (p1,p2,n3) & crossover (p1,p2,n1,0,0) = crossover (p1,p2,n1) & crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,n3) = crossover (p1,p2,n3) & crossover (p1,p2,n1,0,0) = crossover (p1,p2,n1) & crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,0,0,n3) = crossover (p1,p2,n3) & crossover (p1,p2,n1,0,0) = crossover (p1,p2,n1) & crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2) )
 crossover (p1,p2,0,0,n3) = crossover (p1,(crossover (p2,p1,0,0)),n3) by Th12;
hence  crossover (p1,p2,0,0,n3) = crossover (p1,p2,n3) by Th12; ::_thesis: ( crossover (p1,p2,n1,0,0) = crossover (p1,p2,n1) & crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2) )
 crossover (p1,p2,n1,0,0) = crossover ((crossover (p2,p1,n1)),(crossover (p1,p2,n1)),0) by Th8;
hence  crossover (p1,p2,n1,0,0) = crossover (p1,p2,n1) by Th4; ::_thesis: crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2)
 crossover (p1,p2,0,n2,0) = crossover ((crossover (p2,p1,n2)),(crossover (p1,p2,n2)),0) by Th7;
hence  crossover (p1,p2,0,n2,0) = crossover (p1,p2,n2) by Th4; ::_thesis: verum
end;

theorem :: GENEALG1:17
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,0,0,0) = p2
proof
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,0,0,0) = p2
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,0,0,0) = p2
 crossover (p1,p2,0,0,0) = crossover ((crossover (p1,p2,0,0)),p2,0) by Th12;
hence  crossover (p1,p2,0,0,0) = p2 by Th4; ::_thesis: verum
end;

theorem Th18: :: GENEALG1:18
for n1, n2, n3 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3)
proof
let n1, n2, n3 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3)
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n1 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3)
let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3) )
assume A1: n1 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3)
then  n1 >= len S by Def1;
then A2: n1 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3) = crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n1,n2)),n3) by A1, Th9
.= crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n2)),n3) by A2, Th9 ;
hence  crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3) ; ::_thesis: verum
end;

theorem Th19: :: GENEALG1:19
for n2, n1, n3 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n2 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3)
proof
let n2, n1, n3 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S st n2 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3)
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n2 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3)
let p1, p2 be Individual of S; ::_thesis: ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3) )
assume A1: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3)
then  n2 >= len S by Def1;
then A2: n2 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3) = crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1,n2)),n3) by A1, Th10
.= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n3) by A2, Th10 ;
hence  crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3) ; ::_thesis: verum
end;

theorem Th20: :: GENEALG1:20
for n3, n1, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n3 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n2)
proof
let n3, n1, n2 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S st n3 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n2)
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n3 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n2)
let p1, p2 be Individual of S; ::_thesis: ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n2) )
assume  n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n2)
then  n3 >= len S by Def1;
then A1: n3 >= len (crossover (p1,p2,n1,n2)) by Def1;
 crossover (p1,p2,n1,n2,n3) = crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n3) ;
hence  crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n2) by A1, Th5; ::_thesis: verum
end;

theorem Th21: :: GENEALG1:21
for n1, n2, n3 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3)
proof
let n1, n2, n3 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3)
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3)
let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3) )
assume that
A1: n1 >= len p1 and
A2: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3)
 crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3) by A1, Th18;
hence  crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3) by A2, Th9; ::_thesis: verum
end;

theorem :: GENEALG1:22
for n1, n3, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n3 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2)
proof
let n1, n3, n2 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n3 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2)
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n1 >= len p1 & n3 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2)
let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2) )
assume that
A1: n1 >= len p1 and
A2: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2)
 crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n3) by A1, Th18;
hence  crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2) by A2, Th10; ::_thesis: verum
end;

theorem :: GENEALG1:23
for n2, n3, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n2 >= len p1 & n3 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1)
proof
let n2, n3, n1 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S st n2 >= len p1 & n3 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1)
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n2 >= len p1 & n3 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1)
let p1, p2 be Individual of S; ::_thesis: ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1) )
assume that
A1: n2 >= len p1 and
A2: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1)
 crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3) by A1, Th19;
hence  crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1) by A2, Th10; ::_thesis: verum
end;

theorem :: GENEALG1:24
for n1, n2, n3 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 & n3 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = p1
proof
let n1, n2, n3 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 & n3 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = p1
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 & n3 >= len p1 holds
crossover (p1,p2,n1,n2,n3) = p1
let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3) = p1 )
assume that
A1: ( n1 >= len p1 & n2 >= len p1 ) and
A2: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3) = p1
 crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3) by A1, Th21;
hence  crossover (p1,p2,n1,n2,n3) = p1 by A2, Th5; ::_thesis: verum
end;

theorem Th25: :: GENEALG1:25
for n1, n2, n3 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n1,n3) & crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2) )
proof
let n1, n2, n3 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n1,n3) & crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n1,n3) & crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n1,n3) & crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2) )
set q1 = crossover (p1,p2,n1);
set q2 = crossover (p2,p1,n1);
crossover (p1,p2,n1,n2,n3) = crossover ((crossover (p1,p2,n2,n1)),(crossover (p2,p1,n1,n2)),n3) by Th13
.= crossover ((crossover (p1,p2,n2,n1)),(crossover (p2,p1,n2,n1)),n3) by Th13 ;
hence  crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n2,n1,n3) ; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2)
crossover (p1,p2,n1,n2,n3) = crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n2,n3)
.= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n3,n2) by Th13
.= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n2) ;
hence  crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2) ; ::_thesis: verum
end;

theorem Th26: :: GENEALG1:26
for n1, n2, n3 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3,n1,n2)
proof
let n1, n2, n3 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3,n1,n2)
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3,n1,n2)
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3,n1,n2)
 crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n1,n3,n2) by Th25;
hence  crossover (p1,p2,n1,n2,n3) = crossover (p1,p2,n3,n1,n2) by Th25; ::_thesis: verum
end;

theorem Th27: :: GENEALG1:27
for n1, n3, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3) = crossover (p1,p2,n3) & crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n2) & crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n1) )
proof
let n1, n3, n2 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3) = crossover (p1,p2,n3) & crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n2) & crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n1) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3) = crossover (p1,p2,n3) & crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n2) & crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n1) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,n1,n1,n3) = crossover (p1,p2,n3) & crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n2) & crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n1) )
 crossover (p1,p2,n1,n1,n3) = crossover (p1,(crossover (p2,p1,n1,n1)),n3) by Th12;
hence  crossover (p1,p2,n1,n1,n3) = crossover (p1,p2,n3) by Th12; ::_thesis: ( crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n2) & crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n1) )
crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n1,n1,n2) by Th25
.= crossover (p1,(crossover (p2,p1,n1,n1)),n2) by Th12 ;
hence  crossover (p1,p2,n1,n2,n1) = crossover (p1,p2,n2) by Th12; ::_thesis: crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n1)
thus  crossover (p1,p2,n1,n2,n2) = crossover (p1,p2,n2,n2,n1) by Th26
.= crossover (p1,(crossover (p2,p1,n2,n2)),n1) by Th12
.= crossover (p1,p2,n1) by Th12 ; ::_thesis: verum
end;

begin

theorem Th28: :: GENEALG1:28
for n1, n2, n3, n4 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3,n4) is Individual of S
proof
let n1, n2, n3, n4 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) is Individual of S
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3,n4) is Individual of S
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) is Individual of S
reconsider q1 = crossover (p1,p2,n1,n2,n3), q2 = crossover (p2,p1,n1,n2,n3) as Individual of S ;
 crossover (q1,q2,n4) is Individual of S ;
hence  crossover (p1,p2,n1,n2,n3,n4) is Individual of S ; ::_thesis: verum
end;

definition
let S be Gene-Set;
let p1, p2 be Individual of S;
let n1, n2, n3, n4 be Element of NAT ;
:: original: crossover
redefine func crossover (p1,p2,n1,n2,n3,n4) ->  Individual of S;
correctness
coherence
crossover (p1,p2,n1,n2,n3,n4) is Individual of S;
by Th28;
end;

theorem Th29: :: GENEALG1:29
for n2, n3, n4, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3,n4) = crossover (p2,p1,n2,n3,n4) & crossover (p1,p2,n1,0,n3,n4) = crossover (p2,p1,n1,n3,n4) & crossover (p1,p2,n1,n2,0,n4) = crossover (p2,p1,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) )
proof
let n2, n3, n4, n1 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3,n4) = crossover (p2,p1,n2,n3,n4) & crossover (p1,p2,n1,0,n3,n4) = crossover (p2,p1,n1,n3,n4) & crossover (p1,p2,n1,n2,0,n4) = crossover (p2,p1,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3,n4) = crossover (p2,p1,n2,n3,n4) & crossover (p1,p2,n1,0,n3,n4) = crossover (p2,p1,n1,n3,n4) & crossover (p1,p2,n1,n2,0,n4) = crossover (p2,p1,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,0,n2,n3,n4) = crossover (p2,p1,n2,n3,n4) & crossover (p1,p2,n1,0,n3,n4) = crossover (p2,p1,n1,n3,n4) & crossover (p1,p2,n1,n2,0,n4) = crossover (p2,p1,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) )
crossover (p1,p2,0,n2,n3,n4) = crossover ((crossover (p2,p1,n2,n3)),(crossover (p2,p1,0,n2,n3)),n4) by Th15
.= crossover ((crossover (p2,p1,n2,n3)),(crossover (p1,p2,n2,n3)),n4) by Th15 ;
hence  crossover (p1,p2,0,n2,n3,n4) = crossover (p2,p1,n2,n3,n4) ; ::_thesis: ( crossover (p1,p2,n1,0,n3,n4) = crossover (p2,p1,n1,n3,n4) & crossover (p1,p2,n1,n2,0,n4) = crossover (p2,p1,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) )
crossover (p1,p2,n1,0,n3,n4) = crossover ((crossover (p2,p1,n1,n3)),(crossover (p2,p1,n1,0,n3)),n4) by Th15
.= crossover ((crossover (p2,p1,n1,n3)),(crossover (p1,p2,n1,n3)),n4) by Th15 ;
hence  crossover (p1,p2,n1,0,n3,n4) = crossover (p2,p1,n1,n3,n4) ; ::_thesis: ( crossover (p1,p2,n1,n2,0,n4) = crossover (p2,p1,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) )
crossover (p1,p2,n1,n2,0,n4) = crossover ((crossover (p2,p1,n1,n2)),(crossover (p2,p1,n1,n2,0)),n4) by Th15
.= crossover ((crossover (p2,p1,n1,n2)),(crossover (p1,p2,n1,n2)),n4) by Th15 ;
hence  crossover (p1,p2,n1,n2,0,n4) = crossover (p2,p1,n1,n2,n4) ; ::_thesis: crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3)
 crossover (p1,p2,n1,n2,n3,0) = crossover ((crossover (p1,p2,n1,n2,n3)),(crossover (p2,p1,n1,n2,n3)),0) ;
hence  crossover (p1,p2,n1,n2,n3,0) = crossover (p2,p1,n1,n2,n3) by Th4; ::_thesis: verum
end;

theorem Th30: :: GENEALG1:30
for n3, n4, n2, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,n3,n4) = crossover (p1,p2,n3,n4) & crossover (p1,p2,0,n2,0,n4) = crossover (p1,p2,n2,n4) & crossover (p1,p2,0,n2,n3,0) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,0,n3,0) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,0,0,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,0,0) = crossover (p1,p2,n1,n2) )
proof
let n3, n4, n2, n1 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,n3,n4) = crossover (p1,p2,n3,n4) & crossover (p1,p2,0,n2,0,n4) = crossover (p1,p2,n2,n4) & crossover (p1,p2,0,n2,n3,0) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,0,n3,0) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,0,0,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,0,0) = crossover (p1,p2,n1,n2) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,n3,n4) = crossover (p1,p2,n3,n4) & crossover (p1,p2,0,n2,0,n4) = crossover (p1,p2,n2,n4) & crossover (p1,p2,0,n2,n3,0) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,0,n3,0) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,0,0,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,0,0) = crossover (p1,p2,n1,n2) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,0,0,n3,n4) = crossover (p1,p2,n3,n4) & crossover (p1,p2,0,n2,0,n4) = crossover (p1,p2,n2,n4) & crossover (p1,p2,0,n2,n3,0) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,0,n3,0) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,0,0,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,0,0) = crossover (p1,p2,n1,n2) )
 crossover (p1,p2,0,0,n3,n4) = crossover (p2,p1,0,n3,n4) by Th29;
hence  crossover (p1,p2,0,0,n3,n4) = crossover (p1,p2,n3,n4) by Th15; ::_thesis: ( crossover (p1,p2,0,n2,0,n4) = crossover (p1,p2,n2,n4) & crossover (p1,p2,0,n2,n3,0) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,0,n3,0) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,0,0,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,0,0) = crossover (p1,p2,n1,n2) )
 crossover (p1,p2,0,n2,0,n4) = crossover (p2,p1,n2,0,n4) by Th29;
hence  crossover (p1,p2,0,n2,0,n4) = crossover (p1,p2,n2,n4) by Th15; ::_thesis: ( crossover (p1,p2,0,n2,n3,0) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,0,n3,0) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,0,0,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,0,0) = crossover (p1,p2,n1,n2) )
 crossover (p1,p2,0,n2,n3,0) = crossover (p2,p1,n2,n3,0) by Th29;
hence  crossover (p1,p2,0,n2,n3,0) = crossover (p1,p2,n2,n3) by Th15; ::_thesis: ( crossover (p1,p2,n1,0,n3,0) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,0,0,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,0,0) = crossover (p1,p2,n1,n2) )
 crossover (p1,p2,n1,0,n3,0) = crossover (p2,p1,n1,n3,0) by Th29;
hence  crossover (p1,p2,n1,0,n3,0) = crossover (p1,p2,n1,n3) by Th15; ::_thesis: ( crossover (p1,p2,n1,0,0,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,0,0) = crossover (p1,p2,n1,n2) )
 crossover (p1,p2,n1,0,0,n4) = crossover (p2,p1,n1,0,n4) by Th29;
hence  crossover (p1,p2,n1,0,0,n4) = crossover (p1,p2,n1,n4) by Th15; ::_thesis: crossover (p1,p2,n1,n2,0,0) = crossover (p1,p2,n1,n2)
 crossover (p1,p2,n1,n2,0,0) = crossover (p2,p1,n1,n2,0) by Th29;
hence  crossover (p1,p2,n1,n2,0,0) = crossover (p1,p2,n1,n2) by Th15; ::_thesis: verum
end;

theorem Th31: :: GENEALG1:31
for n1, n2, n3, n4 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,0,0,0) = crossover (p2,p1,n1) & crossover (p1,p2,0,n2,0,0) = crossover (p2,p1,n2) & crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) & crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) )
proof
let n1, n2, n3, n4 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,0,0,0) = crossover (p2,p1,n1) & crossover (p1,p2,0,n2,0,0) = crossover (p2,p1,n2) & crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) & crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,0,0,0) = crossover (p2,p1,n1) & crossover (p1,p2,0,n2,0,0) = crossover (p2,p1,n2) & crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) & crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,n1,0,0,0) = crossover (p2,p1,n1) & crossover (p1,p2,0,n2,0,0) = crossover (p2,p1,n2) & crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) & crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) )
 crossover (p1,p2,n1,0,0,0) = crossover (p1,p2,n1,0) by Th30;
hence  crossover (p1,p2,n1,0,0,0) = crossover (p2,p1,n1) by Th8; ::_thesis: ( crossover (p1,p2,0,n2,0,0) = crossover (p2,p1,n2) & crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) & crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) )
 crossover (p1,p2,0,n2,0,0) = crossover (p1,p2,0,n2) by Th30;
hence  crossover (p1,p2,0,n2,0,0) = crossover (p2,p1,n2) by Th7; ::_thesis: ( crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) & crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) )
 crossover (p1,p2,0,0,n3,0) = crossover (p1,p2,0,n3) by Th30;
hence  crossover (p1,p2,0,0,n3,0) = crossover (p2,p1,n3) by Th7; ::_thesis: crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4)
 crossover (p1,p2,0,0,0,n4) = crossover (p1,p2,0,n4) by Th30;
hence  crossover (p1,p2,0,0,0,n4) = crossover (p2,p1,n4) by Th7; ::_thesis: verum
end;

theorem Th32: :: GENEALG1:32
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,0,0,0,0) = p1
proof
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,0,0,0,0) = p1
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,0,0,0,0) = p1
 crossover (p1,p2,0,0,0,0) = crossover (p2,p1,0) by Th31;
hence  crossover (p1,p2,0,0,0,0) = p1 by Th4; ::_thesis: verum
end;

theorem Th33: :: GENEALG1:33
for n1, n2, n3, n4 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n3) ) )
proof
let n1, n2, n3, n4 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n3) ) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n3) ) )
let p1, p2 be Individual of S; ::_thesis: ( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n3) ) )
A1: ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n3) )
proof
assume  n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n3)
then  n4 >= len S by Def1;
then A2: n4 >= len (crossover (p1,p2,n1,n2,n3)) by Def1;
 crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n1,n2,n3)),(crossover (p2,p1,n1,n2,n3)),n4) ;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n3) by A2, Th5; ::_thesis: verum
end;
A3: ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4) )
proof
assume A4: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4)
then  n2 >= len S by Def1;
then A5: n2 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n2,n3)),n4) by A4, Th19
.= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n4) by A5, Th19 ;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4) ; ::_thesis: verum
end;
A6: ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4) )
proof
assume A7: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4)
then  n3 >= len S by Def1;
then A8: n3 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2,n3)),n4) by A7, Th20
.= crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n4) by A8, Th20 ;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4) ; ::_thesis: verum
end;
 ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) )
proof
assume A9: n1 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4)
then  n1 >= len S by Def1;
then A10: n1 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3,n4) = crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n1,n2,n3)),n4) by A9, Th18
.= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n4) by A10, Th18 ;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) ; ::_thesis: verum
end;
hence  ( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n3) ) ) by A3, A6, A1; ::_thesis: verum
end;

theorem Th34: :: GENEALG1:34
for n1, n2, n3, n4 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4) ) & ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4) ) & ( n1 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3) ) & ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4) ) & ( n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3) ) & ( n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2) ) )
proof
let n1, n2, n3, n4 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4) ) & ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4) ) & ( n1 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3) ) & ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4) ) & ( n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3) ) & ( n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2) ) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4) ) & ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4) ) & ( n1 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3) ) & ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4) ) & ( n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3) ) & ( n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2) ) )
let p1, p2 be Individual of S; ::_thesis: ( ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4) ) & ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4) ) & ( n1 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3) ) & ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4) ) & ( n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3) ) & ( n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2) ) )
A1: ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4) )
proof
assume that
A2: n1 >= len p1 and
A3: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4)
 crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) by A2, Th33;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4) by A3, Th19; ::_thesis: verum
end;
A4: ( n1 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3) )
proof
assume that
A5: n1 >= len p1 and
A6: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3)
 crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) by A5, Th33;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3) by A6, Th20; ::_thesis: verum
end;
A7: ( n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2) )
proof
assume that
A8: n3 >= len p1 and
A9: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2)
 crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4) by A8, Th33;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2) by A9, Th20; ::_thesis: verum
end;
A10: ( n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3) )
proof
assume that
A11: n2 >= len p1 and
A12: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3)
 crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4) by A11, Th33;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3) by A12, Th20; ::_thesis: verum
end;
A13: ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4) )
proof
assume that
A14: n2 >= len p1 and
A15: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4)
 crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4) by A14, Th33;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4) by A15, Th19; ::_thesis: verum
end;
 ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4) )
proof
assume that
A16: n1 >= len p1 and
A17: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4)
 crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4) by A16, Th33;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4) by A17, Th18; ::_thesis: verum
end;
hence  ( ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4) ) & ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4) ) & ( n1 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3) ) & ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4) ) & ( n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3) ) & ( n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2) ) ) by A1, A4, A13, A10, A7; ::_thesis: verum
end;

theorem Th35: :: GENEALG1:35
for n1, n2, n3, n4 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1) ) )
proof
let n1, n2, n3, n4 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 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = 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 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1) ) )
let p1, p2 be Individual of S; ::_thesis: ( ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1) ) )
A1: ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4) )
proof
assume that
A2: ( n1 >= len p1 & n2 >= len p1 ) and
A3: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4)
 crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4) by A2, Th34;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4) by A3, Th9; ::_thesis: verum
end;
A4: ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2) )
proof
assume that
A5: ( n1 >= len p1 & n3 >= len p1 ) and
A6: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2)
crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4) by A5, Th34
.= crossover (p1,p2,n4,n2) by Th13 ;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2) by A6, Th9; ::_thesis: verum
end;
A7: ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1) )
proof
assume that
A8: ( n2 >= len p1 & n3 >= len p1 ) and
A9: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1)
crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4) by A8, Th34
.= crossover (p1,p2,n4,n1) by Th13 ;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1) by A9, Th9; ::_thesis: verum
end;
 ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3) )
proof
assume that
A10: ( n1 >= len p1 & n2 >= len p1 ) and
A11: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3)
crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4) by A10, Th34
.= crossover (p1,p2,n4,n3) by Th13 ;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3) by A11, Th9; ::_thesis: verum
end;
hence  ( ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4) ) & ( n1 >= len p1 & n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3) ) & ( n1 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2) ) & ( n2 >= len p1 & n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1) ) ) by A1, A4, A7; ::_thesis: verum
end;

theorem Th36: :: GENEALG1:36
for n1, n2, n3, n4 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 holds
crossover (p1,p2,n1,n2,n3,n4) = p1
proof
let n1, n2, n3, n4 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 holds
crossover (p1,p2,n1,n2,n3,n4) = p1
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 holds
crossover (p1,p2,n1,n2,n3,n4) = p1
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) = p1 )
assume that
A1: ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 ) and
A2: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4) = p1
 crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4) by A1, Th35;
hence  crossover (p1,p2,n1,n2,n3,n4) = p1 by A2, Th5; ::_thesis: verum
end;

theorem Th37: :: GENEALG1:37
for n1, n2, n3, n4 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( 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
let n1, n2, n3, n4 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) = 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) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( 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) )
let p1, p2 be Individual of S; ::_thesis: ( 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) )
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 ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n3,n4)
.= crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n4,n3) by Th13
.= crossover ((crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n4)),(crossover ((crossover (p2,p1,n1,n2)),(crossover (p1,p2,n1,n2)),n4)),n3) ;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n2,n4,n3) ; ::_thesis: verum
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 ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n1,n4)
.= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n4,n1) by Th13
.= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n1) ;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n3,n4,n1) ; ::_thesis: verum
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 ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n2,n4)
.= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),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  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n4,n3,n2) ; ::_thesis: verum
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 ((crossover (p1,p2,n3,n2)),(crossover (p2,p1,n3,n2)),n1,n4)
.= crossover ((crossover (p1,p2,n3,n2)),(crossover (p2,p1,n3,n2)),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  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4,n3,n2,n1) ; ::_thesis: verum
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 ((crossover (p1,p2,n2,n1)),(crossover (p2,p1,n2,n1)),n3,n4)
.= crossover ((crossover (p1,p2,n2,n1)),(crossover (p2,p1,n2,n1)),n4,n3) by Th13
.= crossover ((crossover (p1,p2,n2,n1,n4)),(crossover (p2,p1,n2,n1,n4)),n3) ;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n1,n4,n3) ; ::_thesis: verum
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 ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n1,n4)
.= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),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  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n4,n2,n3,n1) ; ::_thesis: verum
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 ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n2,n4)
.= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n4,n2) by Th13
.= crossover ((crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n4)),(crossover ((crossover (p2,p1,n1,n3)),(crossover (p1,p2,n1,n3)),n4)),n2) ;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n1,n3,n4,n2) ; ::_thesis: verum
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 ((crossover (p1,p2,n3,n2)),(crossover (p2,p1,n3,n2)),n1,n4)
.= crossover ((crossover (p1,p2,n3,n2)),(crossover (p2,p1,n3,n2)),n4,n1) by Th13
.= crossover ((crossover (p1,p2,n3,n2,n4)),(crossover (p2,p1,n3,n2,n4)),n1) ;
hence  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n2,n4,n1) ; ::_thesis: verum
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 ((crossover (p1,p2,n2,n1)),(crossover (p2,p1,n2,n1)),n3,n4)
.= crossover ((crossover (p1,p2,n2,n1)),(crossover (p2,p1,n2,n1)),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  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4,n1,n3) ; ::_thesis: verum
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 ((crossover (p1,p2,n3,n2)),(crossover (p2,p1,n3,n2)),n1,n4)
.= crossover ((crossover (p1,p2,n3,n2)),(crossover (p2,p1,n3,n2)),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  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4,n2,n1) ; ::_thesis: verum
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 ((crossover (p1,p2,n3,n1)),(crossover (p2,p1,n3,n1)),n2,n4)
.= crossover ((crossover (p1,p2,n3,n1)),(crossover (p2,p1,n3,n1)),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  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n3,n4,n1,n2) ; ::_thesis: verum
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 ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n1,n4)
.= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),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  crossover (p1,p2,n1,n2,n3,n4) = crossover (p1,p2,n2,n4,n3,n1) ; ::_thesis: verum
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  ( 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) ) by A2, A15, A1, A8, A4, A6, A3, A10, A14, A12, A9, A13, A11, A7, A5; ::_thesis: verum
end;

theorem Th38: :: GENEALG1:38
for n1, n3, n4, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n4) = crossover (p1,p2,n3,n4) & crossover (p1,p2,n1,n2,n1,n4) = crossover (p1,p2,n2,n4) & crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
proof
let n1, n3, n4, n2 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n4) = crossover (p1,p2,n3,n4) & crossover (p1,p2,n1,n2,n1,n4) = crossover (p1,p2,n2,n4) & crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n4) = crossover (p1,p2,n3,n4) & crossover (p1,p2,n1,n2,n1,n4) = crossover (p1,p2,n2,n4) & crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,n1,n1,n3,n4) = crossover (p1,p2,n3,n4) & crossover (p1,p2,n1,n2,n1,n4) = crossover (p1,p2,n2,n4) & crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
crossover (p1,p2,n1,n1,n3,n4) = crossover ((crossover (p1,p2,n3)),(crossover (p2,p1,n1,n1,n3)),n4) by Th27
.= crossover ((crossover (p1,p2,n3)),(crossover (p2,p1,n3)),n4) by Th27 ;
hence  crossover (p1,p2,n1,n1,n3,n4) = crossover (p1,p2,n3,n4) ; ::_thesis: ( crossover (p1,p2,n1,n2,n1,n4) = crossover (p1,p2,n2,n4) & crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
crossover (p1,p2,n1,n2,n1,n4) = crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n1,n2,n1)),n4) by Th27
.= crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n2)),n4) by Th27 ;
hence  crossover (p1,p2,n1,n2,n1,n4) = crossover (p1,p2,n2,n4) ; ::_thesis: ( crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n2,n3) & crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n1,n1,n2,n3) by Th37
.= crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n1,n1,n2)),n3) by Th27
.= crossover ((crossover (p1,p2,n2)),(crossover (p2,p1,n2)),n3) by Th27 ;
hence  crossover (p1,p2,n1,n2,n3,n1) = crossover (p1,p2,n2,n3) ; ::_thesis: ( crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) & crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
crossover (p1,p2,n1,n2,n2,n4) = crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1,n2,n2)),n4) by Th27
.= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n4) by Th27 ;
hence  crossover (p1,p2,n1,n2,n2,n4) = crossover (p1,p2,n1,n4) ; ::_thesis: ( crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) & crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) )
crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n2,n2,n3) by Th37
.= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1,n2,n2)),n3) by Th27
.= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n3) by Th27 ;
hence  crossover (p1,p2,n1,n2,n3,n2) = crossover (p1,p2,n1,n3) ; ::_thesis: crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2)
crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n3,n3,n2) by Th37
.= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1,n3,n3)),n2) by Th27
.= crossover ((crossover (p1,p2,n1)),(crossover (p2,p1,n1)),n2) by Th27 ;
hence  crossover (p1,p2,n1,n2,n3,n3) = crossover (p1,p2,n1,n2) ; ::_thesis: verum
end;

theorem :: GENEALG1:39
for n1, n3, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n3) = p1 & crossover (p1,p2,n1,n2,n1,n2) = p1 & crossover (p1,p2,n1,n2,n2,n1) = p1 )
proof
let n1, n3, n2 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n3) = p1 & crossover (p1,p2,n1,n2,n1,n2) = p1 & crossover (p1,p2,n1,n2,n2,n1) = p1 )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n3) = p1 & crossover (p1,p2,n1,n2,n1,n2) = p1 & crossover (p1,p2,n1,n2,n2,n1) = p1 )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,n1,n1,n3,n3) = p1 & crossover (p1,p2,n1,n2,n1,n2) = p1 & crossover (p1,p2,n1,n2,n2,n1) = p1 )
 crossover (p1,p2,n1,n1,n3,n3) = crossover (p1,p2,n3,n3) by Th38;
hence  crossover (p1,p2,n1,n1,n3,n3) = p1 by Th12; ::_thesis: ( crossover (p1,p2,n1,n2,n1,n2) = p1 & crossover (p1,p2,n1,n2,n2,n1) = p1 )
 crossover (p1,p2,n1,n2,n1,n2) = crossover (p1,p2,n2,n2) by Th38;
hence  crossover (p1,p2,n1,n2,n1,n2) = p1 by Th12; ::_thesis: crossover (p1,p2,n1,n2,n2,n1) = p1
 crossover (p1,p2,n1,n2,n2,n1) = crossover (p1,p2,n2,n2) by Th38;
hence  crossover (p1,p2,n1,n2,n2,n1) = p1 by Th12; ::_thesis: verum
end;

begin

theorem Th40: :: GENEALG1:40
for n1, n2, n3, n4, n5 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3,n4,n5) is Individual of S
proof
let n1, n2, n3, n4, n5 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) is Individual of S
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3,n4,n5) is Individual of S
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) is Individual of S
reconsider q1 = crossover (p1,p2,n1,n2,n3,n4), q2 = crossover (p2,p1,n1,n2,n3,n4) as Individual of S ;
 crossover (q1,q2,n5) is Individual of S ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) is Individual of S ; ::_thesis: verum
end;

definition
let S be Gene-Set;
let p1, p2 be Individual of S;
let n1, n2, n3, n4, n5 be Element of NAT ;
:: original: crossover
redefine func crossover (p1,p2,n1,n2,n3,n4,n5) ->  Individual of S;
correctness
coherence
crossover (p1,p2,n1,n2,n3,n4,n5) is Individual of S;
by Th40;
end;

theorem Th41: :: GENEALG1:41
for n2, n3, n4, n5, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3,n4,n5) = crossover (p2,p1,n2,n3,n4,n5) & crossover (p1,p2,n1,0,n3,n4,n5) = crossover (p2,p1,n1,n3,n4,n5) & crossover (p1,p2,n1,n2,0,n4,n5) = crossover (p2,p1,n1,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,0,n5) = crossover (p2,p1,n1,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,0) = crossover (p2,p1,n1,n2,n3,n4) )
proof
let n2, n3, n4, n5, n1 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3,n4,n5) = crossover (p2,p1,n2,n3,n4,n5) & crossover (p1,p2,n1,0,n3,n4,n5) = crossover (p2,p1,n1,n3,n4,n5) & crossover (p1,p2,n1,n2,0,n4,n5) = crossover (p2,p1,n1,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,0,n5) = crossover (p2,p1,n1,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,0) = crossover (p2,p1,n1,n2,n3,n4) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3,n4,n5) = crossover (p2,p1,n2,n3,n4,n5) & crossover (p1,p2,n1,0,n3,n4,n5) = crossover (p2,p1,n1,n3,n4,n5) & crossover (p1,p2,n1,n2,0,n4,n5) = crossover (p2,p1,n1,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,0,n5) = crossover (p2,p1,n1,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,0) = crossover (p2,p1,n1,n2,n3,n4) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,0,n2,n3,n4,n5) = crossover (p2,p1,n2,n3,n4,n5) & crossover (p1,p2,n1,0,n3,n4,n5) = crossover (p2,p1,n1,n3,n4,n5) & crossover (p1,p2,n1,n2,0,n4,n5) = crossover (p2,p1,n1,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,0,n5) = crossover (p2,p1,n1,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,0) = crossover (p2,p1,n1,n2,n3,n4) )
A1: crossover (p1,p2,n1,n2,n3,n4,0) = crossover ((crossover (p1,p2,n1,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),0) ;
A2: crossover (p1,p2,n1,0,n3,n4,n5) = crossover ((crossover (p2,p1,n1,n3,n4)),(crossover (p2,p1,n1,0,n3,n4)),n5) by Th29
.= crossover ((crossover (p2,p1,n1,n3,n4)),(crossover (p1,p2,n1,n3,n4)),n5) by Th29 ;
A3: crossover (p1,p2,n1,n2,n3,0,n5) = crossover ((crossover (p2,p1,n1,n2,n3)),(crossover (p2,p1,n1,n2,n3,0)),n5) by Th29
.= crossover ((crossover (p2,p1,n1,n2,n3)),(crossover (p1,p2,n1,n2,n3)),n5) by Th29 ;
A4: crossover (p1,p2,n1,n2,0,n4,n5) = crossover ((crossover (p2,p1,n1,n2,n4)),(crossover (p2,p1,n1,n2,0,n4)),n5) by Th29
.= crossover ((crossover (p2,p1,n1,n2,n4)),(crossover (p1,p2,n1,n2,n4)),n5) by Th29 ;
crossover (p1,p2,0,n2,n3,n4,n5) = crossover ((crossover (p2,p1,n2,n3,n4)),(crossover (p2,p1,0,n2,n3,n4)),n5) by Th29
.= crossover ((crossover (p2,p1,n2,n3,n4)),(crossover (p1,p2,n2,n3,n4)),n5) by Th29 ;
hence  ( crossover (p1,p2,0,n2,n3,n4,n5) = crossover (p2,p1,n2,n3,n4,n5) & crossover (p1,p2,n1,0,n3,n4,n5) = crossover (p2,p1,n1,n3,n4,n5) & crossover (p1,p2,n1,n2,0,n4,n5) = crossover (p2,p1,n1,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,0,n5) = crossover (p2,p1,n1,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,0) = crossover (p2,p1,n1,n2,n3,n4) ) by A2, A4, A3, A1, Th4; ::_thesis: verum
end;

theorem :: GENEALG1:42
for n3, n4, n5, n2, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) & crossover (p1,p2,0,n2,0,n4,n5) = crossover (p1,p2,n2,n4,n5) & crossover (p1,p2,0,n2,n3,0,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,0,n2,n3,n4,0) = crossover (p1,p2,n2,n3,n4) & crossover (p1,p2,n1,0,0,n4,n5) = crossover (p1,p2,n1,n4,n5) & crossover (p1,p2,n1,0,n3,0,n5) = crossover (p1,p2,n1,n3,n5) & crossover (p1,p2,n1,0,n3,n4,0) = crossover (p1,p2,n1,n3,n4) & crossover (p1,p2,n1,n2,0,0,n5) = crossover (p1,p2,n1,n2,n5) & crossover (p1,p2,n1,n2,0,n4,0) = crossover (p1,p2,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0,0) = crossover (p1,p2,n1,n2,n3) )
proof
let n3, n4, n5, n2, n1 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) & crossover (p1,p2,0,n2,0,n4,n5) = crossover (p1,p2,n2,n4,n5) & crossover (p1,p2,0,n2,n3,0,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,0,n2,n3,n4,0) = crossover (p1,p2,n2,n3,n4) & crossover (p1,p2,n1,0,0,n4,n5) = crossover (p1,p2,n1,n4,n5) & crossover (p1,p2,n1,0,n3,0,n5) = crossover (p1,p2,n1,n3,n5) & crossover (p1,p2,n1,0,n3,n4,0) = crossover (p1,p2,n1,n3,n4) & crossover (p1,p2,n1,n2,0,0,n5) = crossover (p1,p2,n1,n2,n5) & crossover (p1,p2,n1,n2,0,n4,0) = crossover (p1,p2,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0,0) = crossover (p1,p2,n1,n2,n3) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) & crossover (p1,p2,0,n2,0,n4,n5) = crossover (p1,p2,n2,n4,n5) & crossover (p1,p2,0,n2,n3,0,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,0,n2,n3,n4,0) = crossover (p1,p2,n2,n3,n4) & crossover (p1,p2,n1,0,0,n4,n5) = crossover (p1,p2,n1,n4,n5) & crossover (p1,p2,n1,0,n3,0,n5) = crossover (p1,p2,n1,n3,n5) & crossover (p1,p2,n1,0,n3,n4,0) = crossover (p1,p2,n1,n3,n4) & crossover (p1,p2,n1,n2,0,0,n5) = crossover (p1,p2,n1,n2,n5) & crossover (p1,p2,n1,n2,0,n4,0) = crossover (p1,p2,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0,0) = crossover (p1,p2,n1,n2,n3) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,0,0,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) & crossover (p1,p2,0,n2,0,n4,n5) = crossover (p1,p2,n2,n4,n5) & crossover (p1,p2,0,n2,n3,0,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,0,n2,n3,n4,0) = crossover (p1,p2,n2,n3,n4) & crossover (p1,p2,n1,0,0,n4,n5) = crossover (p1,p2,n1,n4,n5) & crossover (p1,p2,n1,0,n3,0,n5) = crossover (p1,p2,n1,n3,n5) & crossover (p1,p2,n1,0,n3,n4,0) = crossover (p1,p2,n1,n3,n4) & crossover (p1,p2,n1,n2,0,0,n5) = crossover (p1,p2,n1,n2,n5) & crossover (p1,p2,n1,n2,0,n4,0) = crossover (p1,p2,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0,0) = crossover (p1,p2,n1,n2,n3) )
A1: ( crossover (p1,p2,0,n2,n3,n4,0) = crossover ((crossover (p1,p2,0,n2,n3,n4)),(crossover (p1,p2,n2,n3,n4)),0) & crossover (p1,p2,n1,0,n3,n4,0) = crossover ((crossover (p1,p2,n1,0,n3,n4)),(crossover (p1,p2,n1,n3,n4)),0) ) by Th29;
A2: ( crossover (p1,p2,n1,n2,0,n4,0) = crossover ((crossover (p1,p2,n1,n2,0,n4)),(crossover (p1,p2,n1,n2,n4)),0) & crossover (p1,p2,n1,n2,n3,0,0) = crossover ((crossover (p1,p2,n1,n2,n3,0)),(crossover (p1,p2,n1,n2,n3)),0) ) by Th29;
A3: crossover (p1,p2,n1,0,n3,0,n5) = crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,0,n3,0)),n5) by Th30
.= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n5) by Th30 ;
A4: crossover (p1,p2,n1,0,0,n4,n5) = crossover ((crossover (p1,p2,n1,n4)),(crossover (p2,p1,n1,0,0,n4)),n5) by Th30
.= crossover ((crossover (p1,p2,n1,n4)),(crossover (p2,p1,n1,n4)),n5) by Th30 ;
A5: crossover (p1,p2,n1,n2,0,0,n5) = crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2,0,0)),n5) by Th30
.= crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n5) by Th30 ;
A6: crossover (p1,p2,0,n2,n3,0,n5) = crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,0,n2,n3,0)),n5) by Th30
.= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n5) by Th30 ;
A7: crossover (p1,p2,0,n2,0,n4,n5) = crossover ((crossover (p1,p2,n2,n4)),(crossover (p2,p1,0,n2,0,n4)),n5) by Th30
.= crossover ((crossover (p1,p2,n2,n4)),(crossover (p2,p1,n2,n4)),n5) by Th30 ;
crossover (p1,p2,0,0,n3,n4,n5) = crossover ((crossover (p1,p2,n3,n4)),(crossover (p2,p1,0,0,n3,n4)),n5) by Th30
.= crossover ((crossover (p1,p2,n3,n4)),(crossover (p2,p1,n3,n4)),n5) by Th30 ;
hence  ( crossover (p1,p2,0,0,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) & crossover (p1,p2,0,n2,0,n4,n5) = crossover (p1,p2,n2,n4,n5) & crossover (p1,p2,0,n2,n3,0,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,0,n2,n3,n4,0) = crossover (p1,p2,n2,n3,n4) & crossover (p1,p2,n1,0,0,n4,n5) = crossover (p1,p2,n1,n4,n5) & crossover (p1,p2,n1,0,n3,0,n5) = crossover (p1,p2,n1,n3,n5) & crossover (p1,p2,n1,0,n3,n4,0) = crossover (p1,p2,n1,n3,n4) & crossover (p1,p2,n1,n2,0,0,n5) = crossover (p1,p2,n1,n2,n5) & crossover (p1,p2,n1,n2,0,n4,0) = crossover (p1,p2,n1,n2,n4) & crossover (p1,p2,n1,n2,n3,0,0) = crossover (p1,p2,n1,n2,n3) ) by A7, A6, A4, A3, A1, A5, A2, Th4; ::_thesis: verum
end;

theorem :: GENEALG1:43
for n4, n5, n3, n2, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,0,n4,n5) = crossover (p2,p1,n4,n5) & crossover (p1,p2,0,0,n3,0,n5) = crossover (p2,p1,n3,n5) & crossover (p1,p2,0,0,n3,n4,0) = crossover (p2,p1,n3,n4) & crossover (p1,p2,0,n2,0,0,n5) = crossover (p2,p1,n2,n5) & crossover (p1,p2,0,n2,0,n4,0) = crossover (p2,p1,n2,n4) & crossover (p1,p2,0,n2,n3,0,0) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,0,0,n5) = crossover (p2,p1,n1,n5) & crossover (p1,p2,n1,0,0,n4,0) = crossover (p2,p1,n1,n4) & crossover (p1,p2,n1,0,n3,0,0) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0,0,0) = crossover (p2,p1,n1,n2) )
proof
let n4, n5, n3, n2, n1 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,0,n4,n5) = crossover (p2,p1,n4,n5) & crossover (p1,p2,0,0,n3,0,n5) = crossover (p2,p1,n3,n5) & crossover (p1,p2,0,0,n3,n4,0) = crossover (p2,p1,n3,n4) & crossover (p1,p2,0,n2,0,0,n5) = crossover (p2,p1,n2,n5) & crossover (p1,p2,0,n2,0,n4,0) = crossover (p2,p1,n2,n4) & crossover (p1,p2,0,n2,n3,0,0) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,0,0,n5) = crossover (p2,p1,n1,n5) & crossover (p1,p2,n1,0,0,n4,0) = crossover (p2,p1,n1,n4) & crossover (p1,p2,n1,0,n3,0,0) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0,0,0) = crossover (p2,p1,n1,n2) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,0,n4,n5) = crossover (p2,p1,n4,n5) & crossover (p1,p2,0,0,n3,0,n5) = crossover (p2,p1,n3,n5) & crossover (p1,p2,0,0,n3,n4,0) = crossover (p2,p1,n3,n4) & crossover (p1,p2,0,n2,0,0,n5) = crossover (p2,p1,n2,n5) & crossover (p1,p2,0,n2,0,n4,0) = crossover (p2,p1,n2,n4) & crossover (p1,p2,0,n2,n3,0,0) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,0,0,n5) = crossover (p2,p1,n1,n5) & crossover (p1,p2,n1,0,0,n4,0) = crossover (p2,p1,n1,n4) & crossover (p1,p2,n1,0,n3,0,0) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0,0,0) = crossover (p2,p1,n1,n2) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,0,0,0,n4,n5) = crossover (p2,p1,n4,n5) & crossover (p1,p2,0,0,n3,0,n5) = crossover (p2,p1,n3,n5) & crossover (p1,p2,0,0,n3,n4,0) = crossover (p2,p1,n3,n4) & crossover (p1,p2,0,n2,0,0,n5) = crossover (p2,p1,n2,n5) & crossover (p1,p2,0,n2,0,n4,0) = crossover (p2,p1,n2,n4) & crossover (p1,p2,0,n2,n3,0,0) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,0,0,n5) = crossover (p2,p1,n1,n5) & crossover (p1,p2,n1,0,0,n4,0) = crossover (p2,p1,n1,n4) & crossover (p1,p2,n1,0,n3,0,0) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0,0,0) = crossover (p2,p1,n1,n2) )
A1: ( crossover (p1,p2,0,0,n3,n4,0) = crossover ((crossover (p1,p2,0,0,n3,n4)),(crossover (p2,p1,n3,n4)),0) & crossover (p1,p2,0,n2,0,n4,0) = crossover ((crossover (p1,p2,0,n2,0,n4)),(crossover (p2,p1,n2,n4)),0) ) by Th30;
A2: ( crossover (p1,p2,0,n2,n3,0,0) = crossover ((crossover (p1,p2,0,n2,n3,0)),(crossover (p2,p1,n2,n3)),0) & crossover (p1,p2,n1,0,0,n4,0) = crossover ((crossover (p1,p2,n1,0,0,n4)),(crossover (p2,p1,n1,n4)),0) ) by Th30;
A3: crossover (p1,p2,0,0,n3,0,n5) = crossover ((crossover (p2,p1,n3)),(crossover (p2,p1,0,0,n3,0)),n5) by Th31
.= crossover ((crossover (p2,p1,n3)),(crossover (p1,p2,n3)),n5) by Th31 ;
A4: ( crossover (p1,p2,n1,0,n3,0,0) = crossover ((crossover (p1,p2,n1,0,n3,0)),(crossover (p2,p1,n1,n3)),0) & crossover (p1,p2,n1,n2,0,0,0) = crossover ((crossover (p1,p2,n1,n2,0,0)),(crossover (p2,p1,n1,n2)),0) ) by Th30;
A5: crossover (p1,p2,n1,0,0,0,n5) = crossover ((crossover (p2,p1,n1)),(crossover (p2,p1,n1,0,0,0)),n5) by Th31
.= crossover ((crossover (p2,p1,n1)),(crossover (p1,p2,n1)),n5) by Th31 ;
A6: crossover (p1,p2,0,n2,0,0,n5) = crossover ((crossover (p2,p1,n2)),(crossover (p2,p1,0,n2,0,0)),n5) by Th31
.= crossover ((crossover (p2,p1,n2)),(crossover (p1,p2,n2)),n5) by Th31 ;
crossover (p1,p2,0,0,0,n4,n5) = crossover ((crossover (p2,p1,n4)),(crossover (p2,p1,0,0,0,n4)),n5) by Th31
.= crossover ((crossover (p2,p1,n4)),(crossover (p1,p2,n4)),n5) by Th31 ;
hence  ( crossover (p1,p2,0,0,0,n4,n5) = crossover (p2,p1,n4,n5) & crossover (p1,p2,0,0,n3,0,n5) = crossover (p2,p1,n3,n5) & crossover (p1,p2,0,0,n3,n4,0) = crossover (p2,p1,n3,n4) & crossover (p1,p2,0,n2,0,0,n5) = crossover (p2,p1,n2,n5) & crossover (p1,p2,0,n2,0,n4,0) = crossover (p2,p1,n2,n4) & crossover (p1,p2,0,n2,n3,0,0) = crossover (p2,p1,n2,n3) & crossover (p1,p2,n1,0,0,0,n5) = crossover (p2,p1,n1,n5) & crossover (p1,p2,n1,0,0,n4,0) = crossover (p2,p1,n1,n4) & crossover (p1,p2,n1,0,n3,0,0) = crossover (p2,p1,n1,n3) & crossover (p1,p2,n1,n2,0,0,0) = crossover (p2,p1,n1,n2) ) by A3, A6, A1, A5, A2, A4, Th4; ::_thesis: verum
end;

theorem :: GENEALG1:44
for n5, n4, n3, n2, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,0,0,n5) = crossover (p1,p2,n5) & crossover (p1,p2,0,0,0,n4,0) = crossover (p1,p2,n4) & crossover (p1,p2,0,0,n3,0,0) = crossover (p1,p2,n3) & crossover (p1,p2,0,n2,0,0,0) = crossover (p1,p2,n2) & crossover (p1,p2,n1,0,0,0,0) = crossover (p1,p2,n1) )
proof
let n5, n4, n3, n2, n1 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,0,0,n5) = crossover (p1,p2,n5) & crossover (p1,p2,0,0,0,n4,0) = crossover (p1,p2,n4) & crossover (p1,p2,0,0,n3,0,0) = crossover (p1,p2,n3) & crossover (p1,p2,0,n2,0,0,0) = crossover (p1,p2,n2) & crossover (p1,p2,n1,0,0,0,0) = crossover (p1,p2,n1) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,0,0,0,0,n5) = crossover (p1,p2,n5) & crossover (p1,p2,0,0,0,n4,0) = crossover (p1,p2,n4) & crossover (p1,p2,0,0,n3,0,0) = crossover (p1,p2,n3) & crossover (p1,p2,0,n2,0,0,0) = crossover (p1,p2,n2) & crossover (p1,p2,n1,0,0,0,0) = crossover (p1,p2,n1) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,0,0,0,0,n5) = crossover (p1,p2,n5) & crossover (p1,p2,0,0,0,n4,0) = crossover (p1,p2,n4) & crossover (p1,p2,0,0,n3,0,0) = crossover (p1,p2,n3) & crossover (p1,p2,0,n2,0,0,0) = crossover (p1,p2,n2) & crossover (p1,p2,n1,0,0,0,0) = crossover (p1,p2,n1) )
A1: ( crossover (p1,p2,0,0,n3,0,0) = crossover ((crossover (p1,p2,0,0,n3,0)),(crossover (p1,p2,n3)),0) & crossover (p1,p2,0,n2,0,0,0) = crossover ((crossover (p1,p2,0,n2,0,0)),(crossover (p1,p2,n2)),0) ) by Th31;
A2: crossover (p1,p2,n1,0,0,0,0) = crossover ((crossover (p1,p2,n1,0,0,0)),(crossover (p1,p2,n1)),0) by Th31;
 ( crossover (p1,p2,0,0,0,0,n5) = crossover (p1,(crossover (p2,p1,0,0,0,0)),n5) & crossover (p1,p2,0,0,0,n4,0) = crossover ((crossover (p1,p2,0,0,0,n4)),(crossover (p1,p2,n4)),0) ) by Th31, Th32;
hence  ( crossover (p1,p2,0,0,0,0,n5) = crossover (p1,p2,n5) & crossover (p1,p2,0,0,0,n4,0) = crossover (p1,p2,n4) & crossover (p1,p2,0,0,n3,0,0) = crossover (p1,p2,n3) & crossover (p1,p2,0,n2,0,0,0) = crossover (p1,p2,n2) & crossover (p1,p2,n1,0,0,0,0) = crossover (p1,p2,n1) ) by A1, A2, Th4, Th32; ::_thesis: verum
end;

theorem :: GENEALG1:45
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,0,0,0,0,0) = p2
proof
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,0,0,0,0,0) = p2
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,0,0,0,0,0) = p2
 crossover (p1,p2,0,0,0,0,0) = crossover ((crossover (p1,p2,0,0,0,0)),p2,0) by Th32;
hence  crossover (p1,p2,0,0,0,0,0) = p2 by Th4; ::_thesis: verum
end;

theorem Th46: :: GENEALG1:46
for n1, n2, n3, n4, n5 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4,n5) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4,n5) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4,n5) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n5) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n4) ) )
proof
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 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4,n5) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4,n5) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4,n5) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n5) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n4) ) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4,n5) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4,n5) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4,n5) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n5) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n4) ) )
let p1, p2 be Individual of S; ::_thesis: ( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4,n5) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4,n5) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4,n5) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n5) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n4) ) )
A1: ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n4) )
proof
assume  n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n4)
then  n5 >= len S by Def1;
then A2: n5 >= len (crossover (p1,p2,n1,n2,n3,n4)) by Def1;
 crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n1,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5) ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n4) by A2, Th5; ::_thesis: verum
end;
A3: ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4,n5) )
proof
assume A4: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4,n5)
then  n2 >= len S by Def1;
then A5: n2 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n1,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A4, Th33
.= crossover ((crossover (p1,p2,n1,n3,n4)),(crossover (p2,p1,n1,n3,n4)),n5) by A5, Th33 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4,n5) ; ::_thesis: verum
end;
A6: ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n5) )
proof
assume A7: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n5)
then  n4 >= len S by Def1;
then A8: n4 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n1,n2,n3)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A7, Th33
.= crossover ((crossover (p1,p2,n1,n2,n3)),(crossover (p2,p1,n1,n2,n3)),n5) by A8, Th33 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n5) ; ::_thesis: verum
end;
A9: ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4,n5) )
proof
assume A10: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4,n5)
then  n3 >= len S by Def1;
then A11: n3 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n1,n2,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A10, Th33
.= crossover ((crossover (p1,p2,n1,n2,n4)),(crossover (p2,p1,n1,n2,n4)),n5) by A11, Th33 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4,n5) ; ::_thesis: verum
end;
 ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4,n5) )
proof
assume A12: n1 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4,n5)
then  n1 >= len S by Def1;
then A13: n1 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A12, Th33
.= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n5) by A13, Th33 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4,n5) ; ::_thesis: verum
end;
hence  ( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4,n5) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4,n5) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4,n5) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n5) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3,n4) ) ) by A3, A9, A6, A1; ::_thesis: verum
end;

theorem :: GENEALG1:47
for n1, n2, n3, n4, n5 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) ) & ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4,n5) ) & ( n1 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n5) ) & ( n1 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4) ) & ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4,n5) ) & ( n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n5) ) & ( n2 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4) ) & ( n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n5) ) & ( n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4) ) & ( n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3) ) )
proof
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 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) ) & ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4,n5) ) & ( n1 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n5) ) & ( n1 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4) ) & ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4,n5) ) & ( n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n5) ) & ( n2 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4) ) & ( n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n5) ) & ( n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4) ) & ( n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3) ) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) ) & ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4,n5) ) & ( n1 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n5) ) & ( n1 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4) ) & ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4,n5) ) & ( n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n5) ) & ( n2 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4) ) & ( n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n5) ) & ( n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4) ) & ( n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3) ) )
let p1, p2 be Individual of S; ::_thesis: ( ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) ) & ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4,n5) ) & ( n1 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n5) ) & ( n1 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4) ) & ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4,n5) ) & ( n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n5) ) & ( n2 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4) ) & ( n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n5) ) & ( n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4) ) & ( n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3) ) )
A1: ( n2 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4) )
proof
assume that
A2: n2 >= len p1 and
A3: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4)
 n5 >= len S by A3, Def1;
then A4: n5 >= len (crossover (p1,p2,n1,n3,n4)) by Def1;
 crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n1,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A2, Th33;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4) by A4, Th5; ::_thesis: verum
end;
A5: ( n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4) )
proof
assume that
A6: n3 >= len p1 and
A7: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4)
 n5 >= len S by A7, Def1;
then A8: n5 >= len (crossover (p1,p2,n1,n2,n4)) by Def1;
 crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n1,n2,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A6, Th33;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4) by A8, Th5; ::_thesis: verum
end;
A9: ( n1 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n5) )
proof
assume that
A10: n1 >= len p1 and
A11: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n5)
 n1 >= len S by A10, Def1;
then A12: n1 >= len p2 by Def1;
 n4 >= len S by A11, Def1;
then A13: n4 >= len p2 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 A10, A11, Th34
.= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n5) by A12, A13, Th34 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n5) ; ::_thesis: verum
end;
A14: ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4,n5) )
proof
assume that
A15: n1 >= len p1 and
A16: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4,n5)
 n1 >= len S by A15, Def1;
then A17: n1 >= len p2 by Def1;
 n3 >= len S by A16, Def1;
then A18: n3 >= len p2 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 A15, A16, Th34
.= crossover ((crossover (p1,p2,n2,n4)),(crossover (p2,p1,n2,n4)),n5) by A17, A18, Th34 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4,n5) ; ::_thesis: verum
end;
A19: ( n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n5) )
proof
assume that
A20: n2 >= len p1 and
A21: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n5)
 n2 >= len S by A20, Def1;
then A22: n2 >= len p2 by Def1;
 n4 >= len S by A21, Def1;
then A23: n4 >= len p2 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 A20, A21, Th34
.= crossover ((crossover (p1,p2,n1,n3)),(crossover (p2,p1,n1,n3)),n5) by A22, A23, Th34 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n5) ; ::_thesis: verum
end;
A24: ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4,n5) )
proof
assume that
A25: n2 >= len p1 and
A26: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4,n5)
 n2 >= len S by A25, Def1;
then A27: n2 >= len p2 by Def1;
 n3 >= len S by A26, Def1;
then A28: n3 >= len p2 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 A25, A26, Th34
.= crossover ((crossover (p1,p2,n1,n4)),(crossover (p2,p1,n1,n4)),n5) by A27, A28, Th34 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4,n5) ; ::_thesis: verum
end;
A29: ( n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n5) )
proof
assume that
A30: n3 >= len p1 and
A31: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n5)
 n3 >= len S by A30, Def1;
then A32: n3 >= len p2 by Def1;
 n4 >= len S by A31, Def1;
then A33: n4 >= len p2 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 A30, A31, Th34
.= crossover ((crossover (p1,p2,n1,n2)),(crossover (p2,p1,n1,n2)),n5) by A32, A33, Th34 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n5) ; ::_thesis: verum
end;
A34: ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) )
proof
assume that
A35: n1 >= len p1 and
A36: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4,n5)
 n1 >= len S by A35, Def1;
then A37: n1 >= len p2 by Def1;
 n2 >= len S by A36, Def1;
then A38: n2 >= len p2 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 A35, A36, Th34
.= crossover ((crossover (p1,p2,n3,n4)),(crossover (p2,p1,n3,n4)),n5) by A37, A38, Th34 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) ; ::_thesis: verum
end;
A39: ( n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3) )
proof
assume that
A40: n4 >= len p1 and
A41: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3)
 n5 >= len S by A41, Def1;
then A42: n5 >= len (crossover (p1,p2,n1,n2,n3)) by Def1;
 crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n1,n2,n3)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A40, Th33;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3) by A42, Th5; ::_thesis: verum
end;
 ( n1 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4) )
proof
assume that
A43: n1 >= len p1 and
A44: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4)
 n5 >= len S by A44, Def1;
then A45: n5 >= len (crossover (p1,p2,n2,n3,n4)) by Def1;
 crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by A43, Th33;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4) by A45, Th5; ::_thesis: verum
end;
hence  ( ( n1 >= len p1 & n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) ) & ( n1 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4,n5) ) & ( n1 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n5) ) & ( n1 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3,n4) ) & ( n2 >= len p1 & n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4,n5) ) & ( n2 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n5) ) & ( n2 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3,n4) ) & ( n3 >= len p1 & n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n5) ) & ( n3 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n4) ) & ( n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2,n3) ) ) by A34, A14, A9, A24, A19, A1, A29, A5, A39; ::_thesis: verum
end;

theorem :: GENEALG1:48
for n1, n2, n3, n4, n5 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( ( 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
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 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) ) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( ( 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) ) )
let p1, p2 be Individual of S; ::_thesis: ( ( 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) ) )
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 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4)
 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  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n4) by A4, Th5; ::_thesis: verum
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 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3)
 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  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n3) by A8, Th5; ::_thesis: verum
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 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n5)
 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  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n5) ; ::_thesis: verum
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 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3)
 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  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n3) by A19, Th5; ::_thesis: verum
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 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n5)
 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  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n5) ; ::_thesis: verum
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 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4,n5)
 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  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4,n5) ; ::_thesis: verum
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 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4)
 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  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n4) by A37, Th5; ::_thesis: verum
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 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n5)
 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  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n5) ; ::_thesis: verum
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 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2)
 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  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n1,n2) by A48, Th5; ::_thesis: verum
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 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4)
 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  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n4) by A51, Th5; ::_thesis: verum
end;
hence  ( ( 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) ) ) by A27, A9, A38, A1, A5, A20, A34, A16, A45; ::_thesis: verum
end;

theorem :: GENEALG1:49
for n1, n2, n3, n4, n5 being Element of NAT
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) ) )
proof
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
end;

theorem :: GENEALG1:50
for n1, n2, n3, n4, n5 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 holds
crossover (p1,p2,n1,n2,n3,n4,n5) = p1
proof
let n1, n2, n3, n4, n5 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 holds
crossover (p1,p2,n1,n2,n3,n4,n5) = p1
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S st n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 holds
crossover (p1,p2,n1,n2,n3,n4,n5) = p1
let p1, p2 be Individual of S; ::_thesis: ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 & n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5) = p1 )
assume that
A1: ( n1 >= len p1 & n2 >= len p1 & n3 >= len p1 & n4 >= len p1 ) and
A2: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5) = p1
 crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,(crossover (p2,p1,n1,n2,n3,n4)),n5) by A1, Th36;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = p1 by A2, Th5; ::_thesis: verum
end;

theorem Th51: :: GENEALG1:51
for n1, n2, n3, n4, n5 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n1,n3,n4,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n2,n1,n4,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4,n2,n3,n1,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5,n2,n3,n4,n1) )
proof
let n1, n2, n3, n4, n5 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) = crossover (p1,p2,n2,n1,n3,n4,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n2,n1,n4,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4,n2,n3,n1,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5,n2,n3,n4,n1) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n1,n3,n4,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n2,n1,n4,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4,n2,n3,n1,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5,n2,n3,n4,n1) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n1,n3,n4,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n2,n1,n4,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4,n2,n3,n1,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5,n2,n3,n4,n1) )
A1: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n3,n2,n1,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by Th37
.= crossover ((crossover (p1,p2,n3,n2,n1,n4)),(crossover (p2,p1,n3,n2,n1,n4)),n5) by Th37 ;
A2: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n4,n2,n3,n1)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by Th37
.= crossover ((crossover (p1,p2,n4,n2,n3,n1)),(crossover (p2,p1,n4,n2,n3,n1)),n5) by Th37 ;
A3: crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5,n2,n3,n4,n1)
proof
set q2 = crossover (p2,p1,n2,n3,n4);
set q1 = crossover (p1,p2,n2,n3,n4);
A4: crossover (p1,p2,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n5) ;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n2,n3,n4,n1)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by Th37
.= crossover ((crossover (p1,p2,n2,n3,n4,n1)),(crossover (p2,p1,n2,n3,n4,n1)),n5) by Th37
.= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n1,n5)
.= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n5,n1) by Th13
.= crossover ((crossover (p1,p2,n5,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4,n5)),n1) by A4, Th37
.= crossover ((crossover (p1,p2,n5,n2,n3,n4)),(crossover (p2,p1,n5,n2,n3,n4)),n1) by Th37 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5,n2,n3,n4,n1) ; ::_thesis: verum
end;
crossover (p1,p2,n1,n2,n3,n4,n5) = crossover ((crossover (p1,p2,n2,n1,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n5) by Th37
.= crossover ((crossover (p1,p2,n2,n1,n3,n4)),(crossover (p2,p1,n2,n1,n3,n4)),n5) by Th37 ;
hence  ( crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n2,n1,n3,n4,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n3,n2,n1,n4,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n4,n2,n3,n1,n5) & crossover (p1,p2,n1,n2,n3,n4,n5) = crossover (p1,p2,n5,n2,n3,n4,n1) ) by A1, A2, A3; ::_thesis: verum
end;

theorem Th52: :: GENEALG1:52
for n1, n3, n4, n5, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) & crossover (p1,p2,n1,n2,n1,n4,n5) = crossover (p1,p2,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,n1,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) )
proof
let n1, n3, n4, n5, n2 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) & crossover (p1,p2,n1,n2,n1,n4,n5) = crossover (p1,p2,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,n1,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) & crossover (p1,p2,n1,n2,n1,n4,n5) = crossover (p1,p2,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,n1,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,n1,n1,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) & crossover (p1,p2,n1,n2,n1,n4,n5) = crossover (p1,p2,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,n1,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) )
set q1 = crossover (p1,p2,n2,n3,n4);
set q2 = crossover (p2,p1,n2,n3,n4);
crossover (p1,p2,n1,n1,n3,n4,n5) = crossover ((crossover (p1,p2,n3,n4)),(crossover (p2,p1,n1,n1,n3,n4)),n5) by Th38
.= crossover ((crossover (p1,p2,n3,n4)),(crossover (p2,p1,n3,n4)),n5) by Th38 ;
hence  crossover (p1,p2,n1,n1,n3,n4,n5) = crossover (p1,p2,n3,n4,n5) ; ::_thesis: ( crossover (p1,p2,n1,n2,n1,n4,n5) = crossover (p1,p2,n2,n4,n5) & crossover (p1,p2,n1,n2,n3,n1,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) )
crossover (p1,p2,n1,n2,n1,n4,n5) = crossover ((crossover (p1,p2,n2,n4)),(crossover (p2,p1,n1,n2,n1,n4)),n5) by Th38
.= crossover ((crossover (p1,p2,n2,n4)),(crossover (p2,p1,n2,n4)),n5) by Th38 ;
hence  crossover (p1,p2,n1,n2,n1,n4,n5) = crossover (p1,p2,n2,n4,n5) ; ::_thesis: ( crossover (p1,p2,n1,n2,n3,n1,n5) = crossover (p1,p2,n2,n3,n5) & crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) )
crossover (p1,p2,n1,n2,n3,n1,n5) = crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n1,n2,n3,n1)),n5) by Th38
.= crossover ((crossover (p1,p2,n2,n3)),(crossover (p2,p1,n2,n3)),n5) by Th38 ;
hence  crossover (p1,p2,n1,n2,n3,n1,n5) = crossover (p1,p2,n2,n3,n5) ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4)
crossover (p1,p2,n1,n2,n3,n4,n1) = crossover ((crossover (p1,p2,n2,n3,n4,n1)),(crossover (p2,p1,n1,n2,n3,n4)),n1) by Th37
.= crossover ((crossover (p1,p2,n2,n3,n4,n1)),(crossover (p2,p1,n2,n3,n4,n1)),n1) by Th37
.= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n1,n1) ;
hence  crossover (p1,p2,n1,n2,n3,n4,n1) = crossover (p1,p2,n2,n3,n4) by Th12; ::_thesis: verum
end;

begin

theorem Th53: :: GENEALG1:53
for n1, n2, n3, n4, n5, n6 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3,n4,n5,n6) is Individual of S
proof
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) is Individual of S
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds crossover (p1,p2,n1,n2,n3,n4,n5,n6) is Individual of S
let p1, p2 be Individual of S; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) is Individual of S
reconsider q1 = crossover (p1,p2,n1,n2,n3,n4,n5), q2 = crossover (p2,p1,n1,n2,n3,n4,n5) as Individual of S ;
 crossover (q1,q2,n6) is Individual of S ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5,n6) is Individual of S ; ::_thesis: verum
end;

definition
let S be Gene-Set;
let p1, p2 be Individual of S;
let n1, n2, n3, n4, n5, n6 be Element of NAT ;
:: original: crossover
redefine func crossover (p1,p2,n1,n2,n3,n4,n5,n6) ->  Individual of S;
correctness
coherence
crossover (p1,p2,n1,n2,n3,n4,n5,n6) is Individual of S;
by Th53;
end;

theorem :: GENEALG1:54
for n2, n3, n4, n5, n6, n1 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3,n4,n5,n6) = crossover (p2,p1,n2,n3,n4,n5,n6) & crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover (p2,p1,n1,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover (p2,p1,n1,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
proof
let n2, n3, n4, n5, n6, n1 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3,n4,n5,n6) = crossover (p2,p1,n2,n3,n4,n5,n6) & crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover (p2,p1,n1,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover (p2,p1,n1,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,0,n2,n3,n4,n5,n6) = crossover (p2,p1,n2,n3,n4,n5,n6) & crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover (p2,p1,n1,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover (p2,p1,n1,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,0,n2,n3,n4,n5,n6) = crossover (p2,p1,n2,n3,n4,n5,n6) & crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover (p2,p1,n1,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover (p2,p1,n1,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
crossover (p1,p2,0,n2,n3,n4,n5,n6) = crossover ((crossover (p2,p1,n2,n3,n4,n5)),(crossover (p2,p1,0,n2,n3,n4,n5)),n6) by Th41
.= crossover ((crossover (p2,p1,n2,n3,n4,n5)),(crossover (p1,p2,n2,n3,n4,n5)),n6) by Th41 ;
hence  crossover (p1,p2,0,n2,n3,n4,n5,n6) = crossover (p2,p1,n2,n3,n4,n5,n6) ; ::_thesis: ( crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover (p2,p1,n1,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover (p2,p1,n1,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover ((crossover (p2,p1,n1,n3,n4,n5)),(crossover (p2,p1,n1,0,n3,n4,n5)),n6) by Th41
.= crossover ((crossover (p2,p1,n1,n3,n4,n5)),(crossover (p1,p2,n1,n3,n4,n5)),n6) by Th41 ;
hence  crossover (p1,p2,n1,0,n3,n4,n5,n6) = crossover (p2,p1,n1,n3,n4,n5,n6) ; ::_thesis: ( crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover (p2,p1,n1,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover ((crossover (p2,p1,n1,n2,n4,n5)),(crossover (p2,p1,n1,n2,0,n4,n5)),n6) by Th41
.= crossover ((crossover (p2,p1,n1,n2,n4,n5)),(crossover (p1,p2,n1,n2,n4,n5)),n6) by Th41 ;
hence  crossover (p1,p2,n1,n2,0,n4,n5,n6) = crossover (p2,p1,n1,n2,n4,n5,n6) ; ::_thesis: ( crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover ((crossover (p2,p1,n1,n2,n3,n5)),(crossover (p2,p1,n1,n2,n3,0,n5)),n6) by Th41
.= crossover ((crossover (p2,p1,n1,n2,n3,n5)),(crossover (p1,p2,n1,n2,n3,n5)),n6) by Th41 ;
hence  crossover (p1,p2,n1,n2,n3,0,n5,n6) = crossover (p2,p1,n1,n2,n3,n5,n6) ; ::_thesis: ( crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5) )
crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover ((crossover (p2,p1,n1,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4,0)),n6) by Th41
.= crossover ((crossover (p2,p1,n1,n2,n3,n4)),(crossover (p1,p2,n1,n2,n3,n4)),n6) by Th41 ;
hence  crossover (p1,p2,n1,n2,n3,n4,0,n6) = crossover (p2,p1,n1,n2,n3,n4,n6) ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover (p2,p1,n1,n2,n3,n4,n5)
thus  crossover (p1,p2,n1,n2,n3,n4,n5,0) = crossover ((crossover (p1,p2,n1,n2,n3,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),0)
.= crossover (p2,p1,n1,n2,n3,n4,n5) by Th4 ; ::_thesis: verum
end;

theorem :: GENEALG1:55
for n1, n2, n3, n4, n5, n6 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) ) & ( n6 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) ) )
proof
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
( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) ) & ( n6 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) ) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) ) & ( n6 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) ) )
let p1, p2 be Individual of S; ::_thesis: ( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) ) & ( n6 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) ) )
A1: ( n6 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) )
proof
assume  n6 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5)
then  n6 >= len S by Def1;
then A2: n6 >= len (crossover (p1,p2,n1,n2,n3,n4,n5)) by Def1;
 crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n1,n2,n3,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) by A2, Th5; ::_thesis: verum
end;
A3: ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) )
proof
assume A4: n2 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6)
then  n2 >= len S by Def1;
then A5: n2 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n1,n3,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by A4, Th46
.= crossover ((crossover (p1,p2,n1,n3,n4,n5)),(crossover (p2,p1,n1,n3,n4,n5)),n6) by A5, Th46 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) ; ::_thesis: verum
end;
A6: ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) )
proof
assume A7: n5 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6)
then  n5 >= len S by Def1;
then A8: n5 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n1,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by A7, Th46
.= crossover ((crossover (p1,p2,n1,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4)),n6) by A8, Th46 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) ; ::_thesis: verum
end;
A9: ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) )
proof
assume A10: n4 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6)
then  n4 >= len S by Def1;
then A11: n4 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n1,n2,n3,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by A10, Th46
.= crossover ((crossover (p1,p2,n1,n2,n3,n5)),(crossover (p2,p1,n1,n2,n3,n5)),n6) by A11, Th46 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) ; ::_thesis: verum
end;
A12: ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) )
proof
assume A13: n3 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6)
then  n3 >= len S by Def1;
then A14: n3 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n1,n2,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by A13, Th46
.= crossover ((crossover (p1,p2,n1,n2,n4,n5)),(crossover (p2,p1,n1,n2,n4,n5)),n6) by A14, Th46 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) ; ::_thesis: verum
end;
 ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) )
proof
assume A15: n1 >= len p1 ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6)
then  n1 >= len S by Def1;
then A16: n1 >= len p2 by Def1;
crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n2,n3,n4,n5)),(crossover (p2,p1,n1,n2,n3,n4,n5)),n6) by A15, Th46
.= crossover ((crossover (p1,p2,n2,n3,n4,n5)),(crossover (p2,p1,n2,n3,n4,n5)),n6) by A16, Th46 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) ; ::_thesis: verum
end;
hence  ( ( n1 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n2,n3,n4,n5,n6) ) & ( n2 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n3,n4,n5,n6) ) & ( n3 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n4,n5,n6) ) & ( n4 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n5,n6) ) & ( n5 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n6) ) & ( n6 >= len p1 implies crossover (p1,p2,n1,n2,n3,n4,n5,n6) = crossover (p1,p2,n1,n2,n3,n4,n5) ) ) by A3, A12, A9, A6, A1; ::_thesis: verum
end;

theorem Th56: :: GENEALG1:56
for n1, n2, n3, n4, n5, n6 being Element of NAT
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) )
proof
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
end;

theorem :: GENEALG1:57
for n1, n3, n4, n5, n6, n2 being Element of NAT
for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n4,n5,n6) = crossover (p1,p2,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover (p1,p2,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover (p1,p2,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) )
proof
let n1, n3, n4, n5, n6, n2 be Element of NAT ; ::_thesis: for S being Gene-Set
for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n4,n5,n6) = crossover (p1,p2,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover (p1,p2,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover (p1,p2,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) )
let S be Gene-Set; ::_thesis: for p1, p2 being Individual of S holds
( crossover (p1,p2,n1,n1,n3,n4,n5,n6) = crossover (p1,p2,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover (p1,p2,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover (p1,p2,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) )
let p1, p2 be Individual of S; ::_thesis: ( crossover (p1,p2,n1,n1,n3,n4,n5,n6) = crossover (p1,p2,n3,n4,n5,n6) & crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover (p1,p2,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover (p1,p2,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) )
crossover (p1,p2,n1,n1,n3,n4,n5,n6) = crossover ((crossover (p1,p2,n3,n4,n5)),(crossover (p2,p1,n1,n1,n3,n4,n5)),n6) by Th52
.= crossover ((crossover (p1,p2,n3,n4,n5)),(crossover (p2,p1,n3,n4,n5)),n6) by Th52 ;
hence  crossover (p1,p2,n1,n1,n3,n4,n5,n6) = crossover (p1,p2,n3,n4,n5,n6) ; ::_thesis: ( crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover (p1,p2,n2,n4,n5,n6) & crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover (p1,p2,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) )
crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover ((crossover (p1,p2,n2,n4,n5)),(crossover (p2,p1,n1,n2,n1,n4,n5)),n6) by Th52
.= crossover ((crossover (p1,p2,n2,n4,n5)),(crossover (p2,p1,n2,n4,n5)),n6) by Th52 ;
hence  crossover (p1,p2,n1,n2,n1,n4,n5,n6) = crossover (p1,p2,n2,n4,n5,n6) ; ::_thesis: ( crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover (p1,p2,n2,n3,n5,n6) & crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) )
crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover ((crossover (p1,p2,n2,n3,n5)),(crossover (p2,p1,n1,n2,n3,n1,n5)),n6) by Th52
.= crossover ((crossover (p1,p2,n2,n3,n5)),(crossover (p2,p1,n2,n3,n5)),n6) by Th52 ;
hence  crossover (p1,p2,n1,n2,n3,n1,n5,n6) = crossover (p1,p2,n2,n3,n5,n6) ; ::_thesis: ( crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) & crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) )
crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4,n1)),n6) by Th52
.= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n6) by Th52 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n1,n6) = crossover (p1,p2,n2,n3,n4,n6) ; ::_thesis: crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5)
crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n5,n2,n3,n4,n1,n1) by Th56
.= crossover (p1,p2,n1,n2,n3,n4,n1,n5) by Th56
.= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n1,n2,n3,n4,n1)),n5) by Th52
.= crossover ((crossover (p1,p2,n2,n3,n4)),(crossover (p2,p1,n2,n3,n4)),n5) by Th52 ;
hence  crossover (p1,p2,n1,n2,n3,n4,n5,n1) = crossover (p1,p2,n2,n3,n4,n5) ; ::_thesis: verum
end;