let v1, v2 be Element of V; :: thesis: ( ex F being FinSequence of the carrier of V st

( F is one-to-one & rng F = Carrier L & v1 = Sum (L (#) F) ) & ex F being FinSequence of the carrier of V st

( F is one-to-one & rng F = Carrier L & v2 = Sum (L (#) F) ) implies v1 = v2 )

given F1 being FinSequence of the carrier of V such that A4: F1 is one-to-one and

A5: rng F1 = Carrier L and

A6: v1 = Sum (L (#) F1) ; :: thesis: ( for F being FinSequence of the carrier of V holds

( not F is one-to-one or not rng F = Carrier L or not v2 = Sum (L (#) F) ) or v1 = v2 )

given F2 being FinSequence of the carrier of V such that A7: F2 is one-to-one and

A8: rng F2 = Carrier L and

A9: v2 = Sum (L (#) F2) ; :: thesis: v1 = v2

defpred S_{1}[ object , object ] means {$2} = F1 " {(F2 . $1)};

A10: len F1 = len F2 by A4, A5, A7, A8, FINSEQ_1:48;

A11: dom F1 = Seg (len F1) by FINSEQ_1:def 3;

A12: dom F2 = Seg (len F2) by FINSEQ_1:def 3;

A13: for x being object st x in dom F1 holds

ex y being object st

( y in dom F1 & S_{1}[x,y] )

A15: for x being object st x in dom F1 holds

S_{1}[x,f . x]
from FUNCT_2:sch 1(A13);

A16: rng f = dom F1

A21: len G1 = len F1 by Def5;

A22: f is one-to-one

A33: dom (L (#) F2) = Seg (len (L (#) F2)) by FINSEQ_1:def 3;

reconsider f = f as Permutation of (dom F1) by A16, A22, FUNCT_2:57;

( dom F1 = Seg (len F1) & dom G1 = Seg (len G1) ) by FINSEQ_1:def 3;

then reconsider f = f as Permutation of (dom G1) by A21;

A34: len (L (#) F2) = len F2 by Def5;

A35: dom G1 = Seg (len G1) by FINSEQ_1:def 3;

( F is one-to-one & rng F = Carrier L & v1 = Sum (L (#) F) ) & ex F being FinSequence of the carrier of V st

( F is one-to-one & rng F = Carrier L & v2 = Sum (L (#) F) ) implies v1 = v2 )

given F1 being FinSequence of the carrier of V such that A4: F1 is one-to-one and

A5: rng F1 = Carrier L and

A6: v1 = Sum (L (#) F1) ; :: thesis: ( for F being FinSequence of the carrier of V holds

( not F is one-to-one or not rng F = Carrier L or not v2 = Sum (L (#) F) ) or v1 = v2 )

given F2 being FinSequence of the carrier of V such that A7: F2 is one-to-one and

A8: rng F2 = Carrier L and

A9: v2 = Sum (L (#) F2) ; :: thesis: v1 = v2

defpred S

A10: len F1 = len F2 by A4, A5, A7, A8, FINSEQ_1:48;

A11: dom F1 = Seg (len F1) by FINSEQ_1:def 3;

A12: dom F2 = Seg (len F2) by FINSEQ_1:def 3;

A13: for x being object st x in dom F1 holds

ex y being object st

( y in dom F1 & S

proof

consider f being Function of (dom F1),(dom F1) such that
let x be object ; :: thesis: ( x in dom F1 implies ex y being object st

( y in dom F1 & S_{1}[x,y] ) )

assume x in dom F1 ; :: thesis: ex y being object st

( y in dom F1 & S_{1}[x,y] )

then F2 . x in rng F1 by A5, A8, A10, A11, A12, FUNCT_1:def 3;

then consider y being object such that

A14: F1 " {(F2 . x)} = {y} by A4, FUNCT_1:74;

take y ; :: thesis: ( y in dom F1 & S_{1}[x,y] )

y in F1 " {(F2 . x)} by A14, TARSKI:def 1;

hence y in dom F1 by FUNCT_1:def 7; :: thesis: S_{1}[x,y]

thus S_{1}[x,y]
by A14; :: thesis: verum

end;( y in dom F1 & S

assume x in dom F1 ; :: thesis: ex y being object st

( y in dom F1 & S

then F2 . x in rng F1 by A5, A8, A10, A11, A12, FUNCT_1:def 3;

then consider y being object such that

A14: F1 " {(F2 . x)} = {y} by A4, FUNCT_1:74;

take y ; :: thesis: ( y in dom F1 & S

y in F1 " {(F2 . x)} by A14, TARSKI:def 1;

hence y in dom F1 by FUNCT_1:def 7; :: thesis: S

thus S

A15: for x being object st x in dom F1 holds

S

A16: rng f = dom F1

proof

reconsider G1 = L (#) F1 as FinSequence of V ;
thus
rng f c= dom F1
by RELAT_1:def 19; :: according to XBOOLE_0:def 10 :: thesis: dom F1 c= rng f

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in dom F1 or y in rng f )

assume A17: y in dom F1 ; :: thesis: y in rng f

then F1 . y in rng F2 by A5, A8, FUNCT_1:def 3;

then consider x being object such that

A18: x in dom F2 and

A19: F2 . x = F1 . y by FUNCT_1:def 3;

F1 " {(F2 . x)} = F1 " (Im (F1,y)) by A17, A19, FUNCT_1:59;

then F1 " {(F2 . x)} c= {y} by A4, FUNCT_1:82;

then {(f . x)} c= {y} by A10, A11, A12, A15, A18;

then A20: f . x = y by ZFMISC_1:18;

x in dom f by A10, A11, A12, A18, FUNCT_2:def 1;

hence y in rng f by A20, FUNCT_1:def 3; :: thesis: verum

end;let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in dom F1 or y in rng f )

assume A17: y in dom F1 ; :: thesis: y in rng f

then F1 . y in rng F2 by A5, A8, FUNCT_1:def 3;

then consider x being object such that

A18: x in dom F2 and

A19: F2 . x = F1 . y by FUNCT_1:def 3;

F1 " {(F2 . x)} = F1 " (Im (F1,y)) by A17, A19, FUNCT_1:59;

then F1 " {(F2 . x)} c= {y} by A4, FUNCT_1:82;

then {(f . x)} c= {y} by A10, A11, A12, A15, A18;

then A20: f . x = y by ZFMISC_1:18;

x in dom f by A10, A11, A12, A18, FUNCT_2:def 1;

hence y in rng f by A20, FUNCT_1:def 3; :: thesis: verum

A21: len G1 = len F1 by Def5;

A22: f is one-to-one

proof

set G2 = L (#) F2;
let y1, y2 be object ; :: according to FUNCT_1:def 4 :: thesis: ( not y1 in dom f or not y2 in dom f or not f . y1 = f . y2 or y1 = y2 )

assume that

A23: y1 in dom f and

A24: y2 in dom f and

A25: f . y1 = f . y2 ; :: thesis: y1 = y2

dom F1 = dom f by FUNCT_2:52;

then A26: dom F1 <> {} by A23;

A27: y2 in dom F1 by A26, A24, FUNCT_2:def 1;

then A28: F1 " {(F2 . y2)} = {(f . y2)} by A15;

A29: y1 in dom F1 by A26, A23, FUNCT_2:def 1;

then F2 . y1 in rng F1 by A5, A8, A10, A11, A12, FUNCT_1:def 3;

then A30: {(F2 . y1)} c= rng F1 by ZFMISC_1:31;

F2 . y2 in rng F1 by A5, A8, A10, A11, A12, A27, FUNCT_1:def 3;

then A31: {(F2 . y2)} c= rng F1 by ZFMISC_1:31;

F1 " {(F2 . y1)} = {(f . y1)} by A15, A29;

then {(F2 . y1)} = {(F2 . y2)} by A25, A28, A30, A31, FUNCT_1:91;

then A32: F2 . y1 = F2 . y2 by ZFMISC_1:3;

( y1 in dom F2 & y2 in dom F2 ) by A10, A11, A12, A26, A23, A24, FUNCT_2:def 1;

hence y1 = y2 by A7, A32; :: thesis: verum

end;assume that

A23: y1 in dom f and

A24: y2 in dom f and

A25: f . y1 = f . y2 ; :: thesis: y1 = y2

dom F1 = dom f by FUNCT_2:52;

then A26: dom F1 <> {} by A23;

A27: y2 in dom F1 by A26, A24, FUNCT_2:def 1;

then A28: F1 " {(F2 . y2)} = {(f . y2)} by A15;

A29: y1 in dom F1 by A26, A23, FUNCT_2:def 1;

then F2 . y1 in rng F1 by A5, A8, A10, A11, A12, FUNCT_1:def 3;

then A30: {(F2 . y1)} c= rng F1 by ZFMISC_1:31;

F2 . y2 in rng F1 by A5, A8, A10, A11, A12, A27, FUNCT_1:def 3;

then A31: {(F2 . y2)} c= rng F1 by ZFMISC_1:31;

F1 " {(F2 . y1)} = {(f . y1)} by A15, A29;

then {(F2 . y1)} = {(F2 . y2)} by A25, A28, A30, A31, FUNCT_1:91;

then A32: F2 . y1 = F2 . y2 by ZFMISC_1:3;

( y1 in dom F2 & y2 in dom F2 ) by A10, A11, A12, A26, A23, A24, FUNCT_2:def 1;

hence y1 = y2 by A7, A32; :: thesis: verum

A33: dom (L (#) F2) = Seg (len (L (#) F2)) by FINSEQ_1:def 3;

reconsider f = f as Permutation of (dom F1) by A16, A22, FUNCT_2:57;

( dom F1 = Seg (len F1) & dom G1 = Seg (len G1) ) by FINSEQ_1:def 3;

then reconsider f = f as Permutation of (dom G1) by A21;

A34: len (L (#) F2) = len F2 by Def5;

A35: dom G1 = Seg (len G1) by FINSEQ_1:def 3;

now :: thesis: for i being Nat st i in dom (L (#) F2) holds

(L (#) F2) . i = G1 . (f . i)

hence
v1 = v2
by A1, A4, A5, A6, A7, A8, A9, A21, A34, FINSEQ_1:48, RLVECT_2:6; :: thesis: verum(L (#) F2) . i = G1 . (f . i)

let i be Nat; :: thesis: ( i in dom (L (#) F2) implies (L (#) F2) . i = G1 . (f . i) )

assume A36: i in dom (L (#) F2) ; :: thesis: (L (#) F2) . i = G1 . (f . i)

then A37: ( (L (#) F2) . i = (L . (F2 /. i)) * (F2 /. i) & F2 . i = F2 /. i ) by A34, A12, A33, Def5, PARTFUN1:def 6;

i in dom F2 by A34, A36, FINSEQ_3:29;

then reconsider u = F2 . i as Element of V by FINSEQ_2:11;

i in dom f by A10, A21, A34, A35, A33, A36, FUNCT_2:def 1;

then A38: f . i in dom F1 by A16, FUNCT_1:def 3;

then reconsider m = f . i as Element of NAT by A11;

reconsider v = F1 . m as Element of V by A38, FINSEQ_2:11;

{(F1 . (f . i))} = Im (F1,(f . i)) by A38, FUNCT_1:59

.= F1 .: (F1 " {(F2 . i)}) by A10, A34, A11, A33, A15, A36 ;

then A39: u = v by FUNCT_1:75, ZFMISC_1:18;

F1 . m = F1 /. m by A38, PARTFUN1:def 6;

hence (L (#) F2) . i = G1 . (f . i) by A21, A11, A35, A38, A39, A37, Def5; :: thesis: verum

end;assume A36: i in dom (L (#) F2) ; :: thesis: (L (#) F2) . i = G1 . (f . i)

then A37: ( (L (#) F2) . i = (L . (F2 /. i)) * (F2 /. i) & F2 . i = F2 /. i ) by A34, A12, A33, Def5, PARTFUN1:def 6;

i in dom F2 by A34, A36, FINSEQ_3:29;

then reconsider u = F2 . i as Element of V by FINSEQ_2:11;

i in dom f by A10, A21, A34, A35, A33, A36, FUNCT_2:def 1;

then A38: f . i in dom F1 by A16, FUNCT_1:def 3;

then reconsider m = f . i as Element of NAT by A11;

reconsider v = F1 . m as Element of V by A38, FINSEQ_2:11;

{(F1 . (f . i))} = Im (F1,(f . i)) by A38, FUNCT_1:59

.= F1 .: (F1 " {(F2 . i)}) by A10, A34, A11, A33, A15, A36 ;

then A39: u = v by FUNCT_1:75, ZFMISC_1:18;

F1 . m = F1 /. m by A38, PARTFUN1:def 6;

hence (L (#) F2) . i = G1 . (f . i) by A21, A11, A35, A38, A39, A37, Def5; :: thesis: verum