let o be infinite Ordinal; :: thesis: not LexOrder o is well-ordering
set R = LexOrder o;
set r = RelStr(# (Bags o),(LexOrder o) #);
set ir = the InternalRel of RelStr(# (Bags o),(LexOrder o) #);
set cr = the carrier of RelStr(# (Bags o),(LexOrder o) #);
assume LexOrder o is well-ordering ; :: thesis: contradiction
then A1: LexOrder o is well_founded by WELLORD1:def 4;
the carrier of RelStr(# (Bags o),(LexOrder o) #) = field the InternalRel of RelStr(# (Bags o),(LexOrder o) #) by ORDERS_1:100;
then the InternalRel of RelStr(# (Bags o),(LexOrder o) #) is_well_founded_in the carrier of RelStr(# (Bags o),(LexOrder o) #) by A1, WELLORD1:5;
then A2: RelStr(# (Bags o),(LexOrder o) #) is well_founded by WELLFND1:def 2;
defpred S₁[ set , set ] means $2 = (o --> 0 ) +* $1,1;

A3: now

let n be Element of NAT ; :: thesis: ex y being Element of the carrier of RelStr(# (Bags o),(LexOrder o) #) st S₁[n,y]
set y = (o --> 0 ) +* n,1;
A4: dom (o --> 0 ) = o by FUNCOP_1:19;
reconsider y = (o --> 0 ) +* n,1 as ManySortedSet of ;
A5: ( n in omega & omega c= o ) by CARD_3:102;

now

let x be set ; :: thesis: ( ( x in {n} implies y . x <> 0 ) & ( y . x <> 0 implies x in {n} ) )

hereby :: thesis: ( y . x <> 0 implies x in {n} )

assume x in {n} ; :: thesis: y . x <> 0
then x = n by TARSKI:def 1;
hence y . x <> 0 by A4, A5, FUNCT_7:33; :: thesis: verum

end;

assume that
A6: y . x <> 0 and
A7: not x in {n} ; :: thesis: contradiction
x <> n by A7, TARSKI:def 1;
then A8: y . x = (o --> 0 ) . x by FUNCT_7:34;

per cases ( x in dom (o --> 0 ) or not x in dom (o --> 0 ) ) ;

suppose x in dom (o --> 0 ) ; :: thesis: contradiction

hence contradiction by A6, A8, FUNCOP_1:13; :: thesis: verum

end;

suppose not x in dom (o --> 0 ) ; :: thesis: contradiction

hence contradiction by A6, A8, FUNCT_1:def 4; :: thesis: verum

end;

end;

end;

then support y = {n} by POLYNOM1:def 7;
then y is finite-support by POLYNOM1:def 8;
then reconsider y = y as Element of the carrier of RelStr(# (Bags o),(LexOrder o) #) by POLYNOM1:def 14;
take y = y; :: thesis: S₁[n,y]
thus S₁[n,y] ; :: thesis: verum

end;

consider f being Function of NAT ,the carrier of RelStr(# (Bags o),(LexOrder o) #) such that
A9: for n being Element of NAT holds S₁[n,f . n] from FUNCT_2:sch 3(A3);
reconsider f = f as sequence of RelStr(# (Bags o),(LexOrder o) #) ;
f is descending

proof

let n be Element of NAT ; :: according to WELLFND1:def 7 :: thesis: ( not f . (n + 1) = f . n & [(f . (n + 1)),(f . n)] in the InternalRel of RelStr(# (Bags o),(LexOrder o) #) )
set fn1 = f . (n + 1);
set fn = f . n;
A10: f . (n + 1) = (o --> 0 ) +* (n + 1),1 by A9;
A11: f . n = (o --> 0 ) +* n,1 by A9;
reconsider fn1 = f . (n + 1) as bag of by POLYNOM1:def 14;
reconsider fn = f . n as bag of by POLYNOM1:def 14;
A12: ( n in omega & omega c= o ) by CARD_3:102;
n <> n + 1 ;
then A13: fn1 . n = (o --> 0 ) . n by A10, FUNCT_7:34
.= 0 by A12, FUNCOP_1:13 ;
A14: dom (o --> 0 ) = o by FUNCOP_1:19;
then A15: fn . n = 1 by A11, A12, FUNCT_7:33;

now

let l be Ordinal; :: thesis: ( l in n implies fn1 . l = fn . l )
assume A16: l in n ; :: thesis: fn1 . l = fn . l
then A17: l <> n ;
n < n + 1 by NAT_1:13;
then n in { i where i is Element of NAT : i < n + 1 } ;
then n in n + 1 by AXIOMS:30;
then n c= n + 1 by ORDINAL1:def 2;
then l in n + 1 by A16;
then l <> n + 1 ;
hence fn1 . l = (o --> 0 ) . l by A10, FUNCT_7:34
.= fn . l by A11, A17, FUNCT_7:34 ;
:: thesis: verum

end;

then A18: fn1 < fn by A13, A15, POLYNOM1:def 11;
thus f . (n + 1) <> f . n by A11, A12, A13, A14, FUNCT_7:33; :: thesis: [(f . (n + 1)),(f . n)] in the InternalRel of RelStr(# (Bags o),(LexOrder o) #)
fn1 <=' fn by A18, POLYNOM1:def 12;
hence [(f . (n + 1)),(f . n)] in the InternalRel of RelStr(# (Bags o),(LexOrder o) #) by POLYNOM1:def 16; :: thesis: verum

end;

hence contradiction by A2, WELLFND1:15; :: thesis: verum