let E1, E2 be odd-valued a_partition of n; :: thesis: ( ( for j being Nat
for O being odd-valued FinSequence
for a being natural-valued FinSequence st len O = len p & len p = len a & p = O (#) (2 |^ a) holds
for sort being DoubleReorganization of dom p st 1 = O . (sort _ (1,1)) & ... & 1 = O . (sort _ (1,(len (sort . 1)))) & 3 = O . (sort _ (2,1)) & ... & 3 = O . (sort _ (2,(len (sort . 2)))) & 5 = O . (sort _ (3,1)) & ... & 5 = O . (sort _ (3,(len (sort . 3)))) & ( for i being Nat holds (2 * i) - 1 = O . (sort _ (i,1)) & ... & (2 * i) - 1 = O . (sort _ (i,(len (sort . i)))) ) holds
( card (Coim (E1,1)) = ((2 |^ a) . (sort _ (1,1))) + (((((2 |^ a) *. sort) . 1),2) +...) & card (Coim (E1,3)) = ((2 |^ a) . (sort _ (2,1))) + (((((2 |^ a) *. sort) . 2),2) +...) & card (Coim (E1,5)) = ((2 |^ a) . (sort _ (3,1))) + (((((2 |^ a) *. sort) . 3),2) +...) & card (Coim (E1,((j * 2) - 1))) = ((2 |^ a) . (sort _ (j,1))) + (((((2 |^ a) *. sort) . j),2) +...) ) ) & ( for j being Nat
for O being odd-valued FinSequence
for a being natural-valued FinSequence st len O = len p & len p = len a & p = O (#) (2 |^ a) holds
for sort being DoubleReorganization of dom p st 1 = O . (sort _ (1,1)) & ... & 1 = O . (sort _ (1,(len (sort . 1)))) & 3 = O . (sort _ (2,1)) & ... & 3 = O . (sort _ (2,(len (sort . 2)))) & 5 = O . (sort _ (3,1)) & ... & 5 = O . (sort _ (3,(len (sort . 3)))) & ( for i being Nat holds (2 * i) - 1 = O . (sort _ (i,1)) & ... & (2 * i) - 1 = O . (sort _ (i,(len (sort . i)))) ) holds
( card (Coim (E2,1)) = ((2 |^ a) . (sort _ (1,1))) + (((((2 |^ a) *. sort) . 1),2) +...) & card (Coim (E2,3)) = ((2 |^ a) . (sort _ (2,1))) + (((((2 |^ a) *. sort) . 2),2) +...) & card (Coim (E2,5)) = ((2 |^ a) . (sort _ (3,1))) + (((((2 |^ a) *. sort) . 3),2) +...) & card (Coim (E2,((j * 2) - 1))) = ((2 |^ a) . (sort _ (j,1))) + (((((2 |^ a) *. sort) . j),2) +...) ) ) implies E1 = E2 )
assume that
A1: for j being Nat
for O being odd-valued FinSequence
for a being natural-valued FinSequence st len O = len p & len p = len a & p = O (#) (2 |^ a) holds
for sort being DoubleReorganization of dom p st 1 = O . (sort _ (1,1)) & ... & 1 = O . (sort _ (1,(len (sort . 1)))) & 3 = O . (sort _ (2,1)) & ... & 3 = O . (sort _ (2,(len (sort . 2)))) & 5 = O . (sort _ (3,1)) & ... & 5 = O . (sort _ (3,(len (sort . 3)))) & ( for i being Nat holds (2 * i) - 1 = O . (sort _ (i,1)) & ... & (2 * i) - 1 = O . (sort _ (i,(len (sort . i)))) ) holds
( card (Coim (E1,1)) = ((2 |^ a) . (sort _ (1,1))) + (((((2 |^ a) *. sort) . 1),2) +...) & card (Coim (E1,3)) = ((2 |^ a) . (sort _ (2,1))) + (((((2 |^ a) *. sort) . 2),2) +...) & card (Coim (E1,5)) = ((2 |^ a) . (sort _ (3,1))) + (((((2 |^ a) *. sort) . 3),2) +...) & card (Coim (E1,((j * 2) - 1))) = ((2 |^ a) . (sort _ (j,1))) + (((((2 |^ a) *. sort) . j),2) +...) ) and
A2: for j being Nat
for O being odd-valued FinSequence
for a being natural-valued FinSequence st len O = len p & len p = len a & p = O (#) (2 |^ a) holds
for sort being DoubleReorganization of dom p st 1 = O . (sort _ (1,1)) & ... & 1 = O . (sort _ (1,(len (sort . 1)))) & 3 = O . (sort _ (2,1)) & ... & 3 = O . (sort _ (2,(len (sort . 2)))) & 5 = O . (sort _ (3,1)) & ... & 5 = O . (sort _ (3,(len (sort . 3)))) & ( for i being Nat holds (2 * i) - 1 = O . (sort _ (i,1)) & ... & (2 * i) - 1 = O . (sort _ (i,(len (sort . i)))) ) holds
( card (Coim (E2,1)) = ((2 |^ a) . (sort _ (1,1))) + (((((2 |^ a) *. sort) . 1),2) +...) & card (Coim (E2,3)) = ((2 |^ a) . (sort _ (2,1))) + (((((2 |^ a) *. sort) . 2),2) +...) & card (Coim (E2,5)) = ((2 |^ a) . (sort _ (3,1))) + (((((2 |^ a) *. sort) . 3),2) +...) & card (Coim (E2,((j * 2) - 1))) = ((2 |^ a) . (sort _ (j,1))) + (((((2 |^ a) *. sort) . j),2) +...) ) ; :: thesis: E1 = E2
consider O being odd-valued FinSequence, a being natural-valued FinSequence such that
A3: ( len O = len p & len p = len a & p = O (#) (2 |^ a) ) and
p . 1 = (O . 1) * (2 |^ (a . 1)) & ... & p . (len p) = (O . (len p)) * (2 |^ (a . (len p))) by Th6;
reconsider sort = the odd_organization of O as DoubleReorganization of dom p by A3, FINSEQ_3:29;
A4: (2 * 1) - 1 = 1 ;
A5: (2 * 2) - 1 = 3 ;
(2 * 3) - 1 = 5 ;
then A6: ( 1 = O . (sort _ (1,1)) & ... & 1 = O . (sort _ (1,(len (sort . 1)))) & 3 = O . (sort _ (2,1)) & ... & 3 = O . (sort _ (2,(len (sort . 2)))) & 5 = O . (sort _ (3,1)) & ... & 5 = O . (sort _ (3,(len (sort . 3)))) & ( for i being Nat holds (2 * i) - 1 = O . (sort _ (i,1)) & ... & (2 * i) - 1 = O . (sort _ (i,(len (sort . i)))) ) ) by A4, Def4, A5;
A7: for x being object holds card (Coim (E1,x)) = card (Coim (E2,x))

proof

let x be object ; :: thesis: card (Coim (E1,x)) = card (Coim (E2,x))

per cases ( not x in OddNAT or x in OddNAT ) ;

suppose A8: not x in OddNAT ; :: thesis: card (Coim (E1,x)) = card (Coim (E2,x))

( rng E1 c= OddNAT & rng E2 c= OddNAT ) by Def1;
then ( E1 " {x} = {} & E2 " {x} = {} ) by A8, FUNCT_1:72;
then ( Coim (E1,x) = {} & Coim (E2,x) = {} ) by RELAT_1:def 17;
hence card (Coim (E1,x)) = card (Coim (E2,x)) ; :: thesis: verum

end;

suppose x in OddNAT ; :: thesis: card (Coim (E1,x)) = card (Coim (E2,x))

then consider i being Element of NAT such that
A9: x = (2 * i) + 1 by FIB_NUM2:def 4;
set i1 = i + 1;
card (Coim (E1,(((i + 1) * 2) - 1))) = ((2 |^ a) . (sort _ ((i + 1),1))) + (((((2 |^ a) *. sort) . (i + 1)),2) +...) by A1, A3, A6
.= card (Coim (E2,(((i + 1) * 2) - 1))) by A6, A3, A2 ;
hence card (Coim (E1,x)) = card (Coim (E2,x)) by A9; :: thesis: verum

end;

end;

end;

thus E1 = E2 by RFINSEQ2:19, A7, CLASSES1:def 10; :: thesis: verum